# A HST/WFC3 Thermal Emission Spectrum of the Hot Jupiter HAT-P-7b

Megan Mansfield  Department of Geophysical Sciences, University of Chicago,
5734 S. Ellis Avenue, Chicago, IL 60637, USA  Jacob L. Bean  Department of
Astronomy & Astrophysics, University of Chicago, 5640 S. Ellis Avenue,
Chicago, IL 60637, USA  Michael R. Line  School of Earth and Space
Exploration, Arizona State University, Tempe, AZ 85281, USA  Vivien Parmentier
Aix Marseille Université, CNRS, LAM, Laboratoire d’Astrophysique de Marseille,
Marseille, France  Laura Kreidberg  Harvard-Smithsonian Center for
Astrophysics, Harvard University, Cambridge, MA 02138, USA  Jean-Michel Désert
Anton Pannekoek Institute for Astronomy, University of Amsterdam, 1090 GE
Amsterdam, Netherlands  Jonathan J. Fortney  Department of Astronomy and
Astrophysics, University of California, Santa Cruz, CA 95064, USA  Kevin B.
Stevenson  Space Telescope Science Institute, Baltimore, MD 21218, USA  Jacob
Arcangeli  Anton Pannekoek Institute for Astronomy, University of Amsterdam,
1090 GE Amsterdam, Netherlands  Diana Dragomir  Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Hubble Fellow

###### Abstract

Secondary eclipse observations of several of the hottest hot Jupiters show
featureless, blackbody-like spectra or molecular emission features, which are
consistent with thermal inversions being present in those atmospheres. Theory
predicts a transition between warmer atmospheres with thermal inversions and
cooler atmospheres without inversions, but the exact transition point is
unknown. In order to further investigate this issue, we observed two secondary
eclipses of the hot Jupiter HAT-P-7b with the Hubble Space Telescope (HST)
WFC3 instrument and combined these data with previous Spitzer and Kepler
secondary eclipse observations. The HST and Spitzer data can be well fit by a
blackbody with $T=2692\pm 14$ K, and the Kepler data point constrains the
geometric albedo to $A_{g}=0.077\pm 0.006$. We modeled these data with a 3D
GCM and 1D self-consistent forward models. The 1D models indicate that the
atmosphere has a thermal inversion, weak heat redistribution, and water
dissociation that limits the range of pressures probed. This result suggests
that WFC3 observations of HAT-P-7b and possibly some other ultra-hot Jupiters
appear blackbody-like because they probe a region near the tropopause where
the atmospheric temperature changes slowly with pressure. Additionally, the 1D
models constrain the atmospheric metallicity
($[\text{M/H}]=-0.87^{+0.38}_{-0.34}$) and the carbon-to-oxygen ratio (C/O
$<1$ at 99% confidence). The solar composition 3D GCM matches the Spitzer data
but generally underpredicts the flux in the WFC3 bandpass and cannot reproduce
its featureless shape. This discrepancy could be explained by high atmospheric
drag or nightside clouds, and may be better understood through further
observation with the James Webb Space Telescope (JWST).

planets and satellites: atmospheres - planets and satellites: gaseous planets
- planets and satellites: individual (HAT-P-7b)

††journal: AAS Journals

##  1 Introduction

Thermal emission measurements have revealed a wealth of information about the
compositions and climates of hot Jupiter atmospheres. Composition
determinations for these planets are important because they provide records of
their formation and migration (Venturini et al., 2016; Madhusudhan et al.,
2017). In particular, the core accretion model of planet formation predicts a
trend of decreasing atmospheric metallicity with increasing planet mass for
giant planets (Fortney et al., 2013). Additionally, the carbon-to-oxygen
abundance ratio (C/O) provides information on the environment in which a
planet forms and how the planet migrated to its current location (Öberg et
al., 2011; Madhusudhan et al., 2014; Mordasini et al., 2016; Espinoza et al.,
2017; Ali-Dib, 2017).

Beyond constraints on its composition, the thermal structure of a hot Jupiter
provides information on its climate (Showman & Guillot, 2002; Burrows et al.,
2006). These highly irradiated, tidally locked planets are in a regime
different from any observed in the Solar System, and so exhibit unique
phenomenon, such as high temperatures, large day-night temperature
differences, and high wind speeds (Showman & Guillot, 2002). Additionally,
molecules like TiO or VO can absorb incoming stellar flux and heat the upper
atmosphere, creating a thermal inversion (Hubeny et al., 2003).

Hot Jupiter atmospheres are expected to show a variety of thermal structures
ranging from warmer to cooler planets (Fortney et al., 2008). From an
observational standpoint, planets with cooler dayside temperatures
($T\lessapprox 2000$ K), such as HD 189733b, HD 209458b, and WASP-43b, show
molecular absorption bands in their emission spectra, indicating that they
have temperature-pressure (T-P) profiles exhibiting decreasing temperature
with increasing altitude (Grillmair et al., 2008; Line et al., 2016; Kreidberg
et al., 2014a; Stevenson et al., 2014c). However, the emission spectra of
hotter planets ($T\gtrapprox 2000$ K) are inconsistent with such T-P profiles.
HST/WFC3 observations of WASP-12b and WASP-103b from 1.1-1.7 $\mu$m have
revealed blackbody-like emission spectra, which could be indicative of
isothermal T-P profiles (Swain et al., 2013; Cartier et al., 2017). In
addition, the planets WASP-33b and WASP-121b show signs of molecular emission
bands, which indicate a thermal inversion (Haynes et al., 2015; Evans et al.,
2017). Complicating the picture of a possible smooth transition from non-
inverted to inverted thermal structures is the result for Kepler-13Ab, which
shows molecular absorption features indicative of a non-inverted profile,
possibly because of cold trapping in its atmosphere (Beatty et al., 2017).
WASP-12b also shows signs of molecular absorption in the Spitzer bandpass,
despite its blackbody-like spectrum between 1.1-1.7 $\mu$m (Madhusudhan et
al., 2011; Stevenson et al., 2014b).

In this paper we present the secondary eclipse spectrum of HAT-P-7b observed
with HST/WFC3 in the 1.1-1.7 $\mu$m range. HAT-P-7b is a hot Jupiter with an
intermediate dayside equilibrium temperature of 2600 K (assuming zero albedo
and dayside only recirculation). HAT-P-7b was previously observed with Spitzer
and found to have a thermal inversion (Christiansen et al., 2010; Wong et al.,
2016). However, because Spitzer can only observe exoplanet spectra in a few
broadband regions, there are degeneracies with the molecular abundances that
make it difficult to determine the exact thermal structure with these data
alone (Madhusudhan & Seager, 2010; Line et al., 2014). Spectroscopy in
general, and WFC3 measurements in particular, can remove these degeneracies by
resolving molecular bands. We describe our HST observations of HAT-P-7b in §2.
In §3, we describe our data analysis and results, and we summarize our
findings in §4.

##  2 Observations

We observed secondary eclipses of HAT-P-7b on 23 December 2016 and 4 January
2017 using the HST WFC3 IR detector as part of program GO-14792. We used the
G141 grism to observe the emission spectrum of HAT-P-7b between 1.1-1.7
$\mu$m. Each of the two visits consisted of five consecutive HST orbits, in
which HAT-P-7 was visible for approximately 50 minutes per orbit and occulted
by the Earth for the remainder of each orbit. At the beginning of each orbit,
we took a direct image of the target with the F126N narrow-band filter for
wavelength calibration.

The observations were taken in spatial scan mode with the $256\times 256$
subarray using the SPARS10, NSAMP=16 readout pattern, resulting in a total
exposure time of 103.129 s. We used a scan rate of 0.08 arcsec/second, which
produced spectra extending approximately 80 pixels in the spatial direction
and with peak pixel counts of about 35,000 electrons per pixel. We used bi-
directional scans to maximize the duty cycle, which yielded 21 exposures per
orbit and a duty cycle of 64%. An example spatial scan is shown in Figure 1.
Although the spectrum of a background star overlaps with that of HAT-P-7 in
the full image, our data reduction used the individual ramp samples, in which
the two stars are well separated.

Figure 1:  An example spatial scan. The spectrum of HAT-P-7 overlaps with a
background star in the full image here, but our data reduction used the
individual ramp samples, in which the two stars are well separated.

We reduced the HST data using the data reduction pipeline described in
Kreidberg et al. (2014b). All observation times were converted to
BJD${}_{\text{TDB}}$ (Eastman et al., 2010). We used an optimal extraction
procedure instead of aperture extraction to extract the data (Horne, 1986). We
tested several different aperture sizes to determine one that captured the
full spectrum without capturing large areas of background. We masked cosmic
rays so that optimal extraction could fit the point spread function of the
data without being influenced by cosmic rays. A typical frame had one cosmic
ray masked out. To subtract the background out of each frame, we visually
inspected the images to find a clear background spot on the detector and
subtracted the median of this background area from each pixel in the aperture.
To determine the uncertainties on the measurements we added the photon noise,
read noise, and median absolute deviation of the background in quadrature.

Following standard procedure for HST WFC3 eclipse observations, we discarded
the first orbit of each visit. We also discarded the first exposure from each
orbit in the first visit and the first two exposures from each orbit in the
second visit to improve the quality of the fit. The spectra were binned into
14 channels with a width of 9 pixels per channel, giving a resolution of
$R=30-44$. We also created a broadband white light curve by summing the
spectra over the entire wavelength range.

We fit both the white light curve and the spectroscopic light curves with a
model that combined a secondary eclipse model (Kreidberg, 2015) and a
systematics model based on Berta et al. (2012). For the white light curve, the
free parameters in the secondary eclipse model were the mid-eclipse time
$T_{sec}$ and the planet-to-star flux ratio $F_{p}/F_{s}$. The orbital period,
ratio of the semi-major axis to the stellar radius, inclination, planet-to-
star radius ratio, and eccentricity were fixed to the values determined by
Wong et al. (2016), which were $P=2.2047372$ days, $a/R_{s}=4.03$,
$i=82.2\degree$, $R_{p}/R_{s}=0.07809$, and $e=0.0016$, respectively.

We fit the instrument systematics with an equation of the form

$M(t)=E(t)(cs+vt_{vis})(1-e^{-r_{1}t_{orb}-r_{2}})$ (1)

where $M(t)$ is the modeled flux, $E(t)$ is the eclipse model, $c$ is a
normalization constant, $s$ is a scaling factor to correct for an offset in
normalization between scan directions (McCullough & MacKenty, 2012), $v$ is a
visit-long linear slope, $t_{vis}$ is the time since the beginning of the
visit, $r_{1}$ is the amplitude of an orbit-long exponential ramp, $r_{2}$ is
the time constant of the orbit-long ramp, and $t_{orb}$ is the time since the
beginning of the orbit. $T_{sec}$, $f_{p}/f_{s}$, $r_{1}$, and $r_{2}$ were
fixed to the same values for both visits, while $c$, $v$, and $s$ varied
between visits. The fit to the white light curve thus contained a total of 10
free parameters.

Previous studies of HST WFC3 data have shown that adding a quadratic term to
the visit-long trend in the model of the instrument systematics provides a
better fit, primarily for very bright stars (Stevenson et al., 2014a; Line et
al., 2016). We tested adding a quadratic term to the visit-long trend, but
found Bayesian Information Criterion (BIC) values that were higher by about 8
on average for the quadratic model compared to the linear model, indicating
that the linear model is preferred for this data set.

We estimated the parameters with a Markov Chain Monte Carlo (MCMC) fit using
the emcee package for Python (Foreman-Mackey et al., 2013). The best-fit white
light curve, which is shown in Figure 2, had $\chi^{2}_{\nu}=2.24$ and an
average residual of 90 ppm. The value of $T_{sec}$ determined from the fit to
the white light curve was $2457757.68242\pm 0.00097$ BJD${}_{\text{TDB}}$. The
spectroscopic light curves were fit with the same model as the white light
curve, with the exception that $T_{sec}$ was fixed to the best-fit value from
the fit to the white light curve. An example pairs plot for the MCMC fit to
the 1.234 - 1.271 $\mu$m light curve is shown in Figure 3. The spectroscopic
light curves achieved photon-limited precision, with $\chi^{2}_{\nu}$ values
between $0.89-1.25$. The final secondary eclipse spectrum, along with Spitzer
data from Wong et al. (2016), is shown in Figure 4, and the planet-to-star
flux ratio for each bandpass is listed in Table 1.

Figure 2:  Raw flux of the secondary eclipse of HAT-P-7b (upper panel), best fit broadband white light curve (middle panel), and residuals to the fit (lower panel). Blue points are from the eclipse on 23 December 2016, and green points are from 4 January 2017. The offsets between the two sets of green and blue points in the upper panel are due to the difference between the forward and reverse scans. The fit has $\chi^{2}_{\nu}=2.24$ and an average residual of 90 ppm. Figure 3:  Pairs plot showing the distributions of fit parameters for the MCMC fit to the 1.234 - 1.271 $\mu$m light curve. The off-diagonal panels show marginalized posterior probability for pairs of parameters, with 1, 2, and 3$\sigma$ intervals indicated with black contours. The grey shading is darker for higher probability density. The panels on the diagonal show marginalized posterior probability distributions for each parameter, and the dashed lines indicate the median values and 68% confidence intervals. The planet-to-star flux ratio is not strongly correlated with any of the other fit parameters. For parameters that are allowed to vary between visits ($c$, $v$, and $s$), the distributions are for the eclipse observed on 23 December 2016. Figure 4:  Secondary eclipse spectrum with a suite of theoretical models. Black points with $1\sigma$ error bars represent observations by WFC3 in this paper and by Spitzer in Wong et al. (2016). The insert shows the WFC3 data from 1.1-1.7 $\mu$m. The dark blue line represents the best-fitting 1D atmospheric model, as described in §3.2 , and the surrounding red lines show 500 spectra randomly drawn from the posterior. Blue points outlined in black show the best-fitting 1D model binned to the data resolution. The black line shows a fit to a simple model of thermal and reflected light, as described in §3.3. The orange line shows a model atmosphere with a monotonically decreasing temperature-pressure profile that provides a reasonable match to the Spitzer data, calculated using the methods of Fortney et al. (2008). Table 1: Secondary eclipse spectrum of HAT-P-7b. Wavelength Range ($\mu$m) |  $f_{p}/f_{s}$ (%)
---|---
1.120 - 1.158 | $0.0334\pm 0.0037$
1.158 - 1.196 | $0.0413\pm 0.0038$
1.196 - 1.234 | $0.0404\pm 0.0037$
1.234 - 1.271 | $0.0501\pm 0.0037$
1.271 - 1.309 | $0.0503\pm 0.0038$
1.309 - 1.347 | $0.0498\pm 0.0037$
1.347 - 1.385 | $0.0530\pm 0.0037$
1.385 - 1.423 | $0.0510\pm 0.0037$
1.423 - 1.461 | $0.0547\pm 0.0039$
1.461 - 1.499 | $0.0621\pm 0.0041$
1.499 - 1.536 | $0.0607\pm 0.0042$
1.536 - 1.574 | $0.0593\pm 0.0044$
1.574 - 1.612 | $0.0594\pm 0.0046$
1.612 - 1.650 | $0.0593\pm 0.0045$

##  3 Analysis

The spectrum of HAT-P-7b is shaped like a blackbody and clearly rejects a
model with a monotonically decreasing temperature with altitude, as can be
seen in Figure 4. This could be due to a lack of near-infrared opacity sources
like water in the atmosphere, an isothermal atmospheric structure, or a
previously unrecognized grey opacity obscuring absorption or emission
features. To understand why we see this blackbody-like spectrum, we used three
different modeling approaches: a 3D GCM, 1D self-consistent forward models,
and a simple model of blackbody thermal emission plus reflected stellar light.
The data are well-fit by a blackbody model, so we use these models with
varying amounts of complexity to explore the planetary physics and chemistry
and put the blackbody-like spectrum in context, rather than using fit quality
metrics to search for a single best-fit model.

###  3.1 Fit to 3D GCM

We first performed 3D GCM calculations to help interpret the spectrum and
guide how we approached fitting the data with parameterized models. To model
the three-dimensonal structure of HAT-P-7b, we used the SPARC/MITgcm clobal
circulation model (Showman et al., 2009). Our setup is similar to the one used
in Parmentier et al. (2016) but with planetary and stellar parameters chosen
to match the HAT-P-7 system. The atmospheric opacities and mean molecular
weight used in the calculations correspond to a solar composition atmosphere,
a solar composition atmosphere depleted in TiO and VO, or a solar composition
atmosphere with the abundance of every element apart from hydrogen and helium
increased or decreased by the same amount. We assume local chemical
equilibrium. Atmospheric drag of various possible origins (ohmic dissipation,
hydrodynamic instabilities, etc.) is modeled as a Rayleigh drag present
throughout the whole model and acting with a drag timescale $\tau_{\rm Drag}$.
The model was run for 300 Earth days and all quantities were averaged over the
last 100 days. Figure 5 shows the planet-to-star flux ratio and T-P profile at
the substellar point for each 3D model.

Figure 5:  Planet-to-star flux ratio as a function of wavelength (top) and
dayside temperature-pressure profiles (bottom) for each 3D GCM run. The red,
blue, orange, pink, and magenta lines show models with solar composition,
without TiO and VO, and with metallicities of $+0.5$, $-0.5$, and $-1.0$,
respectively. The dark to light green lines show atmospheres with solar
composition and decreasing drag timescales.

The best fitting 3D models contain both TiO/VO and H- opacities, and have a
thermal inversion due to absorption by TiO/VO in the upper atmosphere. Models
with lower metallicities have a deeper photosphere leading to more efficient
heat redistribution and thus lower fluxes in the WFC3 bandpass. The low-
metallicity models would therefore require even more drag than in the solar
metallicity case to match the observations. Although we include H- opacity, it
does not contribute significantly to the atmospheric opacity. However, water
dissociation has a large impact on the observed spectrum. Water dissociation
limits the range of pressures probed, which limits our observations to the
part of the atmosphere near the tropopause and produces a blackbody-like
spectrum. If water were not dissociating in the upper atmosphere, our
observations would probe a region of the atmosphere which extended above the
tropopause and we would see emission features.

Models containing TiO/VO with varying drag timescales produce similar quality
fits to the Spitzer data points, but the WFC3 points can only be fit well with
a short drag timescale of $\tau_{\rm Drag}=10^{3}$ s. Models with less drag
(i.e. more redistribution) do not produce hot enough daysides to match the
WFC3 data. Preliminary calculations of $\tau_{\rm Drag}$, which scales with
the square of the planetary magnetic field, suggest that such a small
$\tau_{\rm Drag}$ is not unrealistic for such hot planets (Perna et al.,
2010). However, the models with TiO/VO opacities also appear to have small
emission features, which do not match the blackbody-like shape of the WFC3
data. One other possible explanation for the warmer dayside is nightside
clouds, which would increase the greenhouse effect without increasing the
planetary albedo (Stevenson et al., 2014c; Kataria et al., 2015; Parmentier et
al., 2016; Stevenson et al., 2017). Nightside clouds on HAT-P-7b were also
suggested by Armstrong et al. (2016) as an explanation for time variability in
the brightness offset of its phase curve. It remains to be seen whether
Lorentz forces or nightside clouds are the real explanation for why the
nominal GCM underpredicts the WFC3 data.

###  3.2 Fit to 1D grid models

We also retrieve compositional and thermal information by fitting the spectrum
with a grid of self-consistent 1D models. We choose this self-consistent grid-
based approach as spectra with little to no spectral features tend to drive
classic retrievals (e.g., Line et al., 2014) towards unphysical regions of
parameter space.

The 1D models self-consistently solve for the radiative-convective-
thermochemical equillbirium atmosphere solution. For the radiative transfer,
we use the Toon et al. (1989) two stream source function technique to solve
for planetary thermal fluxes at each atmospheric level. Incident stellar flux
at the top of the atmosphere (from a PHOENIX model – Husser et al., 2013) is
treated as a simple exponential attenuation at an average cosine incident
angle of 1/$\sqrt{3}$. The Newton-Raphson iteration technique is used to
determine the layer temperatures that ensure zero net flux divergence across
each model layer. Opacities for H2O, CH4, CO, CO2, NH3, HCN, C2H2, H2S, Na, K,
TiO, VO, FeH, H2-H2/He CIA (Lupu et al., 2014), and H- bound-free and free-
free (John, 1988; Bell & Berrington, 1987) are treated within the “on-the-fly”
correlated-K framework (e.g., Lacis & Oinas, 1991; Amundsen et al., 2016). H2
and He Rayleigh scattering are added in as a continuum absorber. Molecular
abundances are computed using the NASA CEA Gibbs free energy minimization code
(Gordon & Mcbride, 1994) given the Lodders et al. (2009) elemental abundances.

We reduce our parameterization of the 1D atmospheres to 3 parameters: day-
night redistribution (f), which scales the incident stellar flux, a
metallicity, [M/H], which scales the Lodders et al. (2009) elemental
abundances in the CEA routine, and a carbon-to-oxygen ratio (C/O). A grid of
1D models are computed along this 3-vector parameter set with the f ranging
between 1 and 4 in steps of 0.25, [M/H] from -1.5 to 2.5 dex in steps of 0.5
dex, and C/O between 0.1 and 2 on a non-uniform grid that more finely samples
C/O values near 1 (e.g., Mollière et al., 2015). The grid spacing is chosen to
be fine enough that interpolation errors are negligible, and the grid range is
chosen to be broad enough to capture a physically sensible range of values. In
all of these models we assume that the relative abundances of all elements
except C and O remain constant.

To fit the 3-parameter model grid spectra (at an R=100) to the data, we use
the emcee package (Foreman-Mackey et al., 2013) combined with the Python
griddata N-dimensional interpolation routine. Only HST and Spitzer data were
used in the fit. The best-fitting spectrum is shown in Figure 4. We include
uniform prior ranges on $f$, [M/H], and log(C/O) between $1.75-2.66$,
-$1.5-2.5$, and -$1.0-0.3$, respectively. We explored values of $f>2.66$ but
decided to exclude them due to energy balance arguments that values larger
than 2.66 violate energy conservation (e.g., Cowan & Agol, 2011).

Like the 3D GCM, the 1D grid models show a thermal inversion due to TiO/VO
absorption heating the upper atmosphere. Figure 6 shows the T-P profile for
the best-fitting grid model in bold red, with 1$\sigma$ errors representing
the spread in the self-consistent T-P profiles that fall within the posterior
shown by the red shaded area. Thin red and blue lines show the contribution
functions for Spitzer and WFC3 data points, respectively, with dark blue lines
showing data in the water band from 1.33-1.48 $\mu$m. The contribution
functions suggest that the data are primarily probing a region of the
atmosphere near the tropopause where the temperature profile switches from
non-inverted to inverted.

Figure 6:  Temperature-pressure profile for the best-fit 1D interpolated
spectrum (bold red line) and 1$\sigma$ error (red shaded area). Temperatures
are shown on the top x-axis. The model has a thermal inversion. Thin, solid
lines indicate the contribution functions for data points in the Spitzer
bandpass (red), and the WFC3 bandpass (blue), with darker blue lines showing
points inside the water band from 1.33-1.48 $\mu$m. The measurements probe
between about 0.2 bar and 2 mbar, where the T-P profile changes temperature
gradually. The dashed curves show thermo-chemical equilibrium mixing ratios
for important absorbing molecules, computed along the self-consistent T-P
profile shown in bold red. Mixing ratios are shown on the bottom x-axis.

The dashed lines on Figure 6 show the thermo-chemical equilibrium mixing
ratios for a set of important absorbing molecules, computed along the self-
consistent T-P profile for the best fit. The water abundance begins to
decrease rapidly around 2 mbar because water begins to dissociate in the hot
upper atmosphere. This suggests that the WFC3 and Spitzer observations probe
the part of the atmosphere just below the tropopause because water
dissociation limits the pressure range observed in that bandpass. If there
were no water dissociation in the atmosphere, the Spitzer observations would
extend up to higher pressures in the atmosphere and we would observe an
emission feature. However, water is still an important source of molecular
opacity in the WFC3 bandpass, as shown in Figure 7. This figure shows
opacities at a pressure of 0.084 bar, near the part of the atmosphere sampled
by the WFC3 bandpass outside the water band. The opacity from molecular water
dominates over all other opacities at this pressure, including H- opacity, in
contrast to what is found for WASP-18b in Arcangeli et al. (2018).

Figure 7:  Opacities of molecular water (blue), H- (orange), CO (black),
TiO+VO+FeH (red), Na+K (dark blue), and hydrogen/hydrogen and hydrogen/helium
collision-induced absorption (green) at a pressure of 0.084 bar in the best-
fitting 1D model. The red shaded area indicates the WFC3 bandpass, and the
green shaded area indicates the Spitzer bandpass.

Figure 8 shows a pairs plot for the 1D grid model fits. The key ingredient of
this retrieval is the assumption of radiative-convective-thermochemical
equilibrium. The factor $f$ measures heat redistribution, where $f=1$
indicates full redistribution, $f=2$ indicates redistribution over the dayside
only, $f=2.66$ corresponds to zero heat redistribution and is the maximum
possible value (Cowan & Agol, 2011). The best-fitting model has $f=2.56\pm
0.06$, indicating that HAT-P-7b has relatively weak heat redistribution. This
high value of $f$ likely suggests that the thermal emission we observe is
being dominated by a localized “hot spot” on the planetary dayside.

Figure 8:  Pairs plot for the 1D grid models, showing the heat redistribution
($f$), metallicity, and C/O ratio. The off-diagonal panels show marginalized
posterior probability for pairs of parameters, with 1, 2, and 3$\sigma$
intervals indicated with dark red, red, and light red shading. The panels on
the diagonal show marginalized posterior probability distributions for each
parameter, and the dashed lines indicate the median values and 68% confidence
intervals. Blue lines and points indicate values for the best-fit interpolated
spectrum.

Although $f$ is highly correlated with the metallicity, the grid models
produce a strong constraint on $f$. Changing the metallicity affects the value
of $f$ because it changes the observed T-P profile. At higher metallicities,
the abundances of TiO/VO and H2O are both higher, but the abundances of TiO/VO
increase faster than the abundance of H2O. The higher TiO/VO abundances warm
the upper atmosphere due to increased optical absorption, but the deeper
atmosphere probed by WFC3 is cooler in order to maintain radiative
equilibrium, as shown in Figure 9. The cooling of the deeper atmosphere is
then compensated for by increasing $f$, which warms the dayside by
redistributing less of the heat to the nightside. This tradeoff can not
continue indefinitely, however, because at very high metallicities the
spectrum will begin to show emission features due to a stronger thermal
inversion (Figure 9). Similarly, at very low metallicities the spectrum will
begin to show absorption features due to a monotonically decreasing T-P
profile. These changes in the metallicity are degenerate with changes in the
relative abundances of Ti and O, but our model assumes that the ratio of Ti to
O remains constant as the metallicity changes.

Figure 9:  Changes in the emission spectra and T-P profiles of the 1D models
as a function of the metallicity ([M/H]) and heat redistribution ($f$). All of
the T-P profiles and mixing ratios in the left panels are set up in the same
manner as in Figure 6, with the shades of each line corresponding to the dark,
middle, and light spectra shown in the right panels. The top panels show that
increasing both [M/H] and $f$ simultaneously produces the same temperature in
the lower atmosphere. The bottom panels show how at higher [M/H] the inversion
gets stronger and emission features begin to appear. The emission features are
most apparent at high metallicities between $2-4$ $\mu$m.

The models can also constrain the C/O because a higher C/O would decrease the
abundances of TiO and VO in the atmosphere, which would weaken the thermal
inversion. Therefore, the 1D models constrain the C/O $<1$ at 99% confidence.
This continues the trend of planets with C/O $<1$, with no planets having
exotic high C/O values (Benneke, 2015). Decreasing the C/O, however, does not
impact the spectrum shape or T-P profile because the TiO/VO opacity is only
weakly dependent in this part of parameter space. Figure 8 demonstrates that
the grid models only produce an upper limit on the C/O.

###  3.3 Simple Two-Parameter Eclipse Model

Although the previous models only considered thermal emission from the planet,
it has been suggested that reflected starlight may contribute significantly to
the flux observed by HST at near-infrared wavelengths (Schwartz & Cowan, 2015;
Keating & Cowan, 2017). Reflected light can dominate over thermal emission in
the near infrared for hot Jupiters with high albedos. Keating & Cowan (2017)
found that reflected light may contribute significantly to the planetary flux
at secondary eclipse for WASP-43b in the WFC3 bandpass. In order to determine
whether reflected light is important in the WFC3 bandpass for HAT-P-7b, we
performed a simple fit to a model of the combined reflected light and thermal
emission for a planet whose emission can be described as a blackbody and that
has a constant geometric albedo across all wavelengths. For such a planet, the
flux from the planet can be described by

$\frac{F_{p}}{F_{s}}=A_{g}\left(\frac{R_{p}}{a}\right)^{2}+\frac{B_{\lambda}(T_{d})}{F_{s,ph}}\left(\frac{R_{p}}{R_{s}}\right)^{2}$
(2)

where $\frac{F_{p}}{F_{s}}$ is the planet-to-star flux ratio, $A_{g}$ is the
geometric albedo, $R_{p}$ is the planet radius, $R_{s}$ is the stellar radius,
$a$ is the distance between the planet and star, $B_{\lambda}(T_{d})$ is the
blackbody flux at the planetary dayside temperature $T_{d}$, and $F_{s,ph}$ is
the stellar flux in each bandpass (Keating & Cowan, 2017). $F_{s,ph}$ was
determined by interpolating between Phoenix models (Husser et al., 2013) using
the Python package pysynphot to obtain a model with $T=6441$ K, $\log(g)=4.02$
cm s-2, and [Fe/H] $=0.15$, which are the stellar parameters for HAT-P-7
(Torres et al., 2012). For this model, $B_{\lambda}(T_{d})$ and $F_{s,ph}$
were both integrated over the bandpasses of the data.

We fit this model to a combined data set including our WFC3 data and the
Spitzer and Kepler data from Wong et al. (2016) using the emcee Python package
(Foreman-Mackey et al., 2013). The free parameters in this fit were the
geometric albedo $A_{g}$ and the planetary dayside temperature $T_{d}$. This
fit is shown by the green line in Figure 4. We found HAT-P-7b to have a
geometric albedo of $A_{g}=0.077\pm 0.006$ and a dayside temperature of
$T_{d}=2654\pm 17$ K. The albedo and dayside temperature are correlated, but
well constrained by the combined data set because the infrared data, and the
Spitzer data in particular, are primarily sensitive to the temperature, while
the Kepler data add sensitivity to the reflected light. This estimate of the
albedo is fairly consistent with previous estimates of HAT-P-7b’s albedo
(Christiansen et al., 2010; Morris et al., 2013; Wong et al., 2016).

Because the geometric albedo is a function of wavelength, it could be
different in the Kepler and WFC3 bandpasses. To ensure that fitting with a
single geometric albedo was valid, we also performed a fit to the WFC3 and
Spitzer data including only thermal emission. This fit had a best-fit
temperature of $T_{d}=2692\pm 14$ K We found through an F test that including
an albedo parameter does not significantly improve the fit at these
wavelengths longer than $1$ $\mu$m. Therefore, the WFC3 and Spitzer data can
be fit well by a single-temperature blackbody, and the albedo is constrained
primarily by the planet-to-star flux in the Kepler bandpass. Contrary to what
Keating & Cowan (2017) found for WASP-43b, the contribution of reflected light
in the WFC3 bandpass is insignificant for HAT-P-7b.

Although we found that HAT-P-7b has a low albedo, Sudarsky et al. (2000)
suggest that very hot giant planets with effective temperatures over 1500 K
should have high albedos, as they should have silicate clouds forming high in
their atmospheres. For example, MgSiO2 should condense at a pressure of about
0.3 bar (Sudarsky et al., 2000). However, both our 3D and 1D models of
HAT-P-7b predict the presence of TiO/VO high in its atmosphere, at pressures
around 0.1-0.001 bar. The presence of such strong optical absorbers above the
hypothesized silicate cloud deck could explain the low albedo that we observe.

Assuming that the planetary Bond albedo is equal to its geometric albedo, we
can use the simple thermal and reflected light fit to calculate the heat
redistribution across the surface of the planet. The heat redistribution is
given by the equation

$T_{d}=T_{0}(1-A_{B})^{1/4}\left(\frac{2}{3}-\frac{5}{12}\epsilon\right)^{1/4}$
(3)

where $T_{0}=T_{s}\sqrt{\frac{R_{s}}{a}}$ is the irradiation temperature,
$T_{s}$ is the stellar temperature, $A_{B}$ is the Bond albedo, and $\epsilon$
is the redistribution efficiency (Schwartz & Cowan, 2015). $\epsilon$ is the
inverse of the parameter $f$ used in our 1D modeling, so smaller values of
$\epsilon$ indicate less efficient heat redistribution. Using this equation,
we find that the redistribution is $\epsilon=0.38\pm 0.11$, which indicates
that HAT-P-7b has very inefficient heat redistribution.

##  4 Discussion

1D grid modeling of the spectrum of HAT-P-7b suggests that the atmosphere
contains a thermal inversion, and constrains the metallicity
($[\text{M/H}]=-0.87^{+0.38}_{-0.34}$) and carbon-to-oxygen ratio (C/O $<1$ at
99% confidence). The new self-consistent 1D model developed in this paper,
which assumes thermochemical and radiative-convective equilibrium, allows
measurement of the atmospheric metallicity even though no molecular features
are directly observed because of the feedback of metallicity onto the strength
of the inversion. Figure 10 shows the atmospheric metallicity as a function of
planet mass for solar system planets (Wong et al., 2004; Fletcher et al.,
2009; Karkoschka & Tomasko, 2011; Sromovsky et al., 2011) and exoplanets. The
exoplanet measurements plotted in Figure 10 are based on either water
abundances (red points) or the new self-consistent modeling developed in this
paper (blue points). The exoplanet data plotted in this figure, and their
sources, are listed in Table 2. The inferred metallicity of HAT-P-7b is well
below the metallicity predicted by the trend in Figure 10. The low metallicity
of HAT-P-7b could just be due to the expected intrinsic scatter around this
trend (Fortney et al., 2013). Alternatively, Madhusudhan et al. (2014)
predicted that sub-solar oxygen and carbon abundances could indicate a planet
that formed farther away from its star and then underwent disk-free migration.
They also predicted that planets which formed in this manner would have super-
solar C/O ratios. Our upper limit on the C/O for HAT-P-7b does not
definitively reveal whether its C/O ratio is super- or sub-solar, so more
observations will be necessary to determine whether this theory can explain
its low metallicity. This new modeling has the potential to reveal the
compositions of other planets with blackbody-like spectra, such as WASP-12b
(Swain et al., 2013).

Figure 10:  Trend in atmospheric metallicity vs. mass for solar system planets
(black points, Wong et al., 2004; Fletcher et al., 2009; Karkoschka & Tomasko,
2011; Sromovsky et al., 2011) and exoplanets with visible water features (red
points, see Table 2). Blue points show results for planets without clear
molecular detections, for which the metallicities have been determined using
the new self-consistent modeling developed in this paper and in Arcangeli et
al. (2018). The black dotted line is a fit to the values for the solar system
planets, but plateauing at 1 once the planet metallicity equals the stellar
metallicity. Table 2: Exoplanet mass and metallicity data plotted in Figure 10

Name |  Planet Mass (M${}_{\text{jup}})$ |  Source |  Planet Metallicity ($\times$ solar, $1\sigma$ range) |  Source |  Stellar Metallicity ([M/H] or [Fe/H]) |  Source
---|---|---|---|---|---|---
HAT-P-7b${}^{\text{a}}$ |  1.78 |  Pál et al. (2008) |  $0.06-0.3$ |  this work |  $0.15\pm 0.08$ |  Torres et al. (2012)
HAT-P-11b${}^{\text{b}}$ |  0.081 |  Bakos et al. (2010) |  $3.0-300$ |  Fraine et al. (2014) |  $0.33\pm 0.07$ |  Torres et al. (2012)
HAT-P-26b${}^{\text{b}}$ |  0.059 |  Hartman et al. (2011) |  $0.8-26.3$ |  Wakeford et al. (2017a) |  $0.10\pm 0.08$ |  Torres et al. (2012)
HD209458b |  0.69 |  Torres et al. (2008) |  $0.06-9.8^{\text{b}}$ |  Line et al. (2016) |  $0.00\pm 0.05$ |  Torres et al. (2008)
|  |  |  $0.1-1.0^{\text{c}}$ |  Brogi et al. (2017) |  |
WASP-12b${}^{\text{b}}$ |  1.41 |  Hebb et al. (2009) |  $0.3-20.0$ |  Kreidberg et al. (2015) |  $0.07\pm 0.07$ |  Torres et al. (2012)
WASP-18b${}^{\text{a}}$ |  10.2 |  Triaud et al. (2010) |  $0.6-2.0$ |  Arcangeli et al. (2018) |  $0.11\pm 0.07$ |  Torres et al. (2012)
WASP-39b${}^{\text{b}}$ |  0.28 |  Faedi et al. (2011) |  $105-199$ |  Wakeford et al. (2017b) |  $-0.12\pm 0.10$ |  Faedi et al. (2011)
WASP-43b |  2.03 |  Gillon et al. (2012) |  $0.4-1.7^{\text{b}}$ |  Stevenson et al. (2017) |  $-0.05\pm 0.17$ |  Hellier et al. (2011)
|  |  |  $0.3-1.7^{\text{d}}$ |  |  |
WASP-103b${}^{\text{a}}$ |  1.49 |  Gillon et al. (2014) |  $10-53$ |  Kreidberg et al., submitted |  $0.06\pm 0.13$ |  Gillon et al. (2014)

${}^{\text{a}}$From self-consistent modeling

${}^{\text{b}}$From H2O detection

${}^{\text{c}}$From low-resolution + high-resolution spectroscopy

${}^{\text{d}}$From CO + CO2

The 1D modeling also strongly constrains the dayside temperature because the
atmosphere is almost isothermal over the bandpass observed by HST and Spitzer,
and shows that HAT-P-7b has very poor heat redistribution. This agrees with a
simple model of combined emitted and reflected light, which also shows that
HAT-P-7b has a high dayside temperature, weak heat redistribution, and a low
albedo. This is also in agreement with previous Spitzer phase curves of
HAT-P-7b, which showed that the planet had a hot dayside with $T=2667\pm 57$
K, a very low albedo, and weak heat redistribution (Wong et al., 2016). The
weak heat redistribution predicted by these models fits the general trend that
has been observed that planets at higher irradiation temperatures have less
efficient heat redistribution, and therefore warmer daysides (Schwartz &
Cowan, 2015).

Although the 1D modeling indicates that HAT-P-7b has a thermal inversion due
to absorption by TiO/VO, this inversion is not definitively observed because
the data can be well-fit by a blackbody with $T=2692\pm 14$ K. The best-
fitting 1D model suggests that the atmosphere does contain a thermal
inversion, and that the WFC3 and Spitzer data sample a part of the atmosphere
near the tropopause where the T-P profile switches from non-inverted to
inverted. In this region, the contribution functions are wide relative to the
scale of the changes in the T-P profile, so the observations all appear to
probe regions of similar temperatures and produce a featureless spectrum.
However, this case can not be distinguished from a completely isothermal
atmosphere. Even our model does not return a perfectly isothermal atmosphere,
and it is possible that there are spectral features in the WFC3 bandpass below
our level of precision. The WFC3 spectrum of WASP-18b, which has a signal-to-
noise nearly four times that of our spectrum, shows some subtle emission
features, and so a higher-precision spectrum of HAT-P-7b may reveal a similar
structure (Arcangeli et al., 2018).

The 1D modeling also indicates that the blackbody-like spectrum of HAT-P-7b is
produced because we are probing a range of pressures in the atmosphere, all of
which have similar temperatures, and not just because only one pressure level
is probed at all wavelengths. However, water dissociation in the upper
atmosphere does limit the range of pressures probed. Other similar hot
planets, like WASP-18b and WASP-103b, display muted spectral features in the
WFC3 bandpass because of water dissociation and H- opacity (Arcangeli et al.,
2018, Kreidberg et al. submitted, Parmentier et al. submitted).

We also modeled the spectrum of HAT-P-7b with a 3D GCM. However, the GCM is
unable to precisely reproduce the observed spectrum. Both the solar
composition and the low-metallicity GCMs are colder on the dayside than the
data suggest, which could be due to increased Lorentz forces causing more drag
in the atmosphere or nightside clouds warming the planet. Additionally, the
modeled spectra appear to have small emission features, which do not match the
observed blackbody-like spectrum. In order to understand the cause of this
discrepancy between the 3D GCM and the data for very hot planets like
HAT-P-7b, more spectroscopic data are required over wider wavelength ranges
and a larger parameter space exploration of parameters such as metallicity and
Ti/O ratio in the GCM is needed.

Overall, the large dayside temperature of HAT-P-7b is extremely puzzling as
the very low heat redistribution it implies cannot be reproduced by current
GCMs. More observations and theoretical work are needed to understand what
causes such a poor energy redistribution. Additionally, because our
observations did not reveal any spectral features, further observations will
be necessary to confirm the low value of the planet’s metallicity. The James
Webb Space Telescope (JWST) will have the ability to spectroscopically observe
thermal emission of transiting planets over a large wavelength range.
Spectroscopic observations of thermal emission with JWST at the wavelengths
covered by Spitzer, where the GCM models for HAT-P-7b show the largest
differences from each other, could resolve the discrepancy between the GCM and
the observations. High-resolution spectroscopy (Snellen et al., 2010) could
also aid in understanding the thermal structure of HAT-P-7b by directly
detecting the TiO responsible for the thermal inversion, as Nugroho et al.
(2017) did for WASP-33b. Additionally, phase curves taken using JWST or the
Atmospheric Remote-sensing Exoplanet Large-survey (ARIEL, Tinetti et al.,
2016) could determine if nightside clouds are responsible for heating the
dayside to temperatures above those predicted by the GCM.

Support for program GO-14792 was provided by NASA through a grant from the
Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.
J.L.B. acknowledges support from the David and Lucile Packard Foundation.
M.R.L. acknowledges NASA XRP grant NNX17AB56G for partial support of the
theoretical interpretation of the data as well as the ASU Research Computing
staff for support with the Saguaro and Agave computing clusters. J.M.D.
acknowledges that the research leading to these results has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (grant agreement no. 679633; Exo-
Atmos). D.D. acknowledges support provided by NASA through Hubble Fellowship
grant HSTHF2-51372.001-A awarded by the Space Telescope Science Institute.

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# Advantages for controls imposed in a proper subset111This study is partially
supported by the National Natural Science Foundation of China under grants
10871154, 10831007, 10801041, 11161130003 and 11171264; and by the National
Basis Research Program of China (973 Program) under grant 2011CB808002.

Gengsheng Wang 222 School of Mathematics and Statistics, Wuhan University,
Wuhan, 430072, China. E-mail<EMAIL_ADDRESS>, Yashan Xu333 School of
Mathematical Sciences, Fudan University, Shanghai 200433, China. E-mail:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we study time optimal control problems for heat equations on
$\Omega\times\mathbb{R}^{+}$. Two properties under consideration are the
existence and the bang-bang properties of time optimal controls. It is proved
that those two properties hold when controls are imposed on some proper
subsets of $\Omega$; while they do not stand when controls are active on the
whole $\Omega$. Besides, a new property for eigenfunctions of $-\Delta$ with
Dirichlet boundary condition is revealed.

Keywords. time optimal control, heat equations, bang-bang property, property
of eigenfunctions of the Laplacian

2010 MSC. 34H15 49N20

##  1 Introduction.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Let $\omega$ be a non
empty and open subset of $\Omega$. Write $\chi_{\omega}$ for its
characteristic function. Consider the following controlled heat equation:

$\left\\{\begin{array}[]{lll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\partial_{t}y(x,t)-\triangle
y(x,t)=\chi_{\omega}(x)u(x,t)&\mbox{in}&\Omega\times\mathbb{R}^{+},\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr
y(x,t)=0&\mbox{on}&\partial\Omega\times\mathbb{R}^{+},\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr y(x,0)=y_{0}(x)&\mbox{in}&\Omega,\end{array}\right.$
(1.1)

where $u$ is a control function taken from a control constraint set and
$y_{0}$ is an initial state taken from $L^{2}(\Omega)$. The solution of (1.1)
corresponding to $u$ and $y_{0}$ will be treated as a function from
$\mathbb{R}^{+}$ to $L^{2}(\Omega)$ and denoted by $y(\cdot;u,y_{0})$.

The purpose of this study is to reveal the following fact: Some properties
hold for some time optimal control problems of (1.1) when $\omega$ is a proper
subset of $\Omega$, but do not stand when $\omega=\Omega$. Consequently, the
local control may be more effective than the global control for heat equations
in some cases.

We begin with introducing time optimal control problems. Let
$\bigr{\\{}\xi_{i}\bigl{\\}}^{\infty}_{i=1}$ be a complete set of
eigenfunctions for $-\Delta$ with Dirichlet boundary condition such that it
serves as a normalized orthonormal basis of $L^{2}(\Omega)$. Write
$\bigr{\\{}\lambda_{i}\bigl{\\}}^{\infty}_{i=1}$, with
$0<\lambda_{1}<\lambda_{2}\leq\cdots<+\infty$, for the corresponding set of
eigenvalues. Then, we take the following target set:

${\cal
S}_{m}=span\bigr{\\{}\,\xi_{m+1},\xi_{m+2},\cdots\bigl{\\}},\;\;\mbox{where}\;\;m\geq
2\;\;\mbox{is arbitrarily fixed}.$

Next, we define, for each natural number $k$ and each finite sequence of
positive numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$, the following control
constraint set:

${\cal
U}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}=\left\\{\sum\limits_{i=1}^{k}\alpha_{i}(\cdot)\xi_{i}\;\biggm{|}\;\mbox{each}\;\;\alpha_{i}(\cdot)\;\;\mbox{is
measurable
from}\;\;\mathbb{R}^{+}\;\;\mbox{to}\;\;\left[-\bar{a}_{i},\bar{a}_{i}\right]\right\\}.$

Consider the following time optimal control problem:

${({\cal P})}\hskip 28.45274pt\displaystyle\inf\limits\left\\{t\geq
0\;\big{|}\;y(t;u,y_{0})\in{\cal S}_{m}\right\\},\;\;\mbox{where the infimum
is taken over all}\;\;u\in{\cal U}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}.$

Two properties of Problem $(\mathcal{P})$ under consideration are as follows:
$(i)$ The existence of time optimal controls; $(ii)$ The bang-bang property:
any optimal control $u^{*}=\sum_{i=1}^{k}\alpha^{*}_{i}\xi_{i}$ satisfies that
for each $i$, $|\alpha^{*}_{i}(t)|=\bar{a}_{i}$ for almost every
$t\in(0,t^{*})$, where $t^{*}$ is the optimal time. In the case that $\Omega$,
$\omega$, $k$ and $y_{0}\notin\mathcal{S}$ are fixed, we say Problem
$(\mathcal{P})$ has optimal controls if for any finite sequence of positive
numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$, it has optimal controls. When
$\omega=\Omega$, $y_{0}\notin\mathcal{S}_{m}$ and $k$ are given, Problem
$(\mathcal{P})$ has optimal controls if and only if there is a finite sequence
of positive numbers $\\{\bar{b}_{i}\\}_{i=1}^{k}$ such that the problem
$(\mathcal{P})$, with $\\{\bar{b}_{i}\\}_{i=1}^{k}$, has optimal controls (see
Remark 2.3).

The main results of this paper are broadly stated as follows: $(a)$ Suppose
that $\omega=\Omega$ and $y_{0}\notin\mathcal{S}_{m}$. Then, $k$ and $y_{0}$
are such that Problem $({\cal P})$ has no optimal control if and only if $k<m$
and $y_{0}$ satisfies

$\displaystyle\left(\mathop{\langle}y_{0},\xi_{k+1}\mathop{\rangle},\mathop{\langle}y_{0},\xi_{k+2}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{m}\mathop{\rangle}\right)^{T}\neq
0;$ (1.2)

$(b)$ Suppose that $\omega=\Omega$ and $y_{0}\notin\mathcal{S}_{m}$. Assume
that either $k\geq m$ or $k<m$ and $y_{0}$ does not satisfy (1.2). Then, in
general, Problem $(\mathcal{P})$ does not hold the bang-bang property; $(c)$
Suppose that $\Omega$ and $\omega$ satisfy accordingly the following
conditions:

  * •

(D1) The eigenvalues $\lambda_{1}\cdots\lambda_{m}$ are simple, i.e.,
$\lambda_{1}<\lambda_{2}<\cdots<\lambda_{m}$,

and

  * •

(D2) $\mathop{\langle}\chi_{\omega}\xi_{i}\,,~{}\xi_{j}\mathop{\rangle}\neq
0\quad{\rm
for~{}all~{}}i\in\\{1,2,\cdots,m\\}\;\;\mbox{and}\;\;j\in\\{1,2,\cdots,k\\}.$

Then, for each $k\geq 1$, each $y_{0}\notin\mathcal{S}_{m}$ and each finite
sequence of positive numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$, Problem
$(\mathcal{P})$ has optimal controls and holds the bang-bang property.

It is worth mentioning that for any fixed bounded domain $\Omega$, there are a
lot of open subsets $\omega$ in $\Omega$ such that
$\mathop{\langle}\chi_{\omega}\xi_{i}\,,~{}\xi_{j}\mathop{\rangle}\neq 0$ for
all $i,j=1,2,\cdots$ (see Theorem 4.2 for a new property of the eigenfunctions
$\\{\xi_{i}\\}_{i=1}^{\infty}$); while there are a lot of bounded domains
$\Omega$ such that the property $\bf(D1)$ holds (see Remark 4.1)).

##  2 Studies of Problem $(\mathcal{P})$ where $\Omega=\omega$

The following result is another version of Theorem 2.5 in [4]. It will be used
later.

######  Lemma 2.1.

Let $\hat{A}\in\mathbb{R}^{d\times d}$ and $\hat{B}\in\mathbb{R}^{d\times l}$,
where $d$ and $l$ are natural numbers. Suppose that

$rank\left(\hat{B},\hat{A}\hat{B},\hat{A}^{2}\hat{B},\cdots,\hat{A}^{d-1}\hat{B}\right)=d,$
(2.1)

and the spectrum of $\hat{A}$ belongs to the left half plane of $\mathbb{C}$.
Then, for each finite sequence of positive numbers $\\{b_{i}\\}_{i=1}^{l}$ and
each $w_{0}$ in $\mathbb{R}^{d}$, there are a $\hat{t}\geq 0$ and a control
$\hat{\beta}$ in the set:

$\bar{\cal
V}\triangleq\left\\{\beta=\left(\beta_{1},\cdots,\beta_{l}\right)^{T}\bigm{|}\;\mbox{each}\;\;\beta_{i}\;\mbox{is
measurable from}\;\mathbb{R}^{+}\;\;\mbox{to}\;\;[-b_{i},b_{i}]\right\\},$
(2.2)

such that the solution $\hat{w}(\cdot;\hat{\beta},w_{0})$ to the equation:

$\left\\{\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\dot{\hat{w}}(t)=\hat{A}\hat{w}(t)+\hat{B}\hat{\beta}(t),&t\in\mathbb{R}^{+},\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hat{w}(0)=w_{0},\end{array}\right.$
(2.3)

reaches zero at $\hat{t}$.

######  Theorem 2.2.

Suppose $\omega=\Omega$ and $y_{0}\notin{\cal S}_{m}$. Then, $k$ and $y_{0}$
are such that Problem $({\cal P})$ has no optimal control if and only if $k<m$
and $y_{0}$ satisfies (1.2).

Proof. The proof will be organized in three steps as follows:

Step 1. Suppose that $k<m$ and $y_{0}$ satisfies (1.2). Then, for any finite
sequence of positive numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$, Problem $({\cal
P})$ has no optimal control.

Let $\\{\bar{a}_{i}\\}_{i=1}^{k}$ be a finite sequence of positive numbers.
Then each $u(\cdot)\in{\cal U}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}$ can be expressed
as $u(t)=\sum\limits_{i=1}^{k}\alpha_{i}(t)\xi_{i}$. Write
$y(t;u,y_{0})=\sum\limits^{\infty}_{i=1}y_{i}(t)\xi_{i}$. Clearly, the
controlled equation (1.1) is equivalent to the following system:

$\displaystyle\dot{y}_{i}(t)+\lambda_{i}y_{i}(t)=\sum\limits_{j=1}^{k}\alpha_{j}(t)\mathop{\langle}\chi_{\omega}\xi_{i},\xi_{j}\mathop{\rangle},\;\;y_{i}(0)=\left<y_{0},\xi_{i}\right>,\qquad
i=1,2,\cdots.$

Write

$z(t)=\left(\begin{array}[]{c}y_{1}(t)\\\ y_{2}(t)\\\ \cdots\\\
y_{m}(t)\end{array}\right),\quad A=\left(\begin{array}[]{cccc}\lambda_{1}\\\
&\lambda_{2}\\\ &&\ddots\\\
&&&\lambda_{m}\end{array}\right),\quad\alpha(t)=\left(\begin{array}[]{c}\alpha_{1}(t)\\\
\alpha_{2}(t)\\\ \cdots\\\ \alpha_{k}(t)\end{array}\right),$

and

$B=\Bigr{(}\mathop{\langle}\chi_{\omega}\xi_{i}\,,~{}\xi_{j}\mathop{\rangle}\Bigl{)}_{i,j}\in\mathbb{R}^{m\times
k}.$ (2.4)

Let

$U_{\\{\bar{a}_{i}\\}^{k}_{i=1}}=[-\bar{a}_{1},\bar{a}_{1}]\times[-\bar{a}_{2},\bar{a}_{2}]\times\cdots\times[-\bar{a}_{k},\bar{a}_{k}].$

Consider the following time optimal control problem:

$(\widetilde{{\cal P}})\hskip 42.67912pt\displaystyle\inf\limits\left\\{t\geq
0\;\big{|}\;z(t;\alpha,z_{0})=0\right\\},\hskip 99.58464pt$

where the infimum is taken over all $\alpha$ from the control constraint set:

$\displaystyle{\cal
V}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}\triangleq\left\\{\alpha=(\alpha_{1},\cdots,\alpha_{k})^{T}\bigm{|}\;\mbox{each}\;\;\alpha_{i}\;\mbox{is
measurable
from}\;\mathbb{R}^{+}\;\;\mbox{to}\;\;[-\bar{a}_{i},\bar{a}_{i}]\right\\},$

and $z(\cdot;\alpha,z_{0})$ is the solution to the following equation:

$\left\\{\begin{array}[]{lll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\dot{z}(t)+Az(t)=B\alpha(t),\;\;t\in\mathbb{R}^{+},\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr
z(0)=\left(\mathop{\langle}y_{0},\xi_{1}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{m}\mathop{\rangle}\right)^{T}.\end{array}\right.$
(2.5)

Clearly, Problems $(\mathcal{P})$ and $(\widetilde{{\cal P}})$ are equivalent,
i.e., $t^{*}$ and $u^{*}=\sum_{i=1}^{k}\alpha^{*}_{i}\xi_{i}$ are accordingly
the optimal time and an optimal control to Problem $(\mathcal{P})$ if and only
if $t^{*}$ and $(\alpha^{*}_{1},\cdots,\alpha^{*}_{k})^{T}$ are the optimal
time and an optimal control to Problem $(\widetilde{{\cal P}})$ respectively.

Since $\omega=\Omega$ and $k<m$, it follows from (2.4) that
$B=\left(\begin{array}[]{c}I_{k\times k}\\\ 0\end{array}\right)$ in this case.
Let $z_{1}(t)=(y_{1}(t),\cdots,y_{k}(t))^{T}$ and
$z_{2}(t)=(y_{k+1}(t),\cdots,y_{m}(t))^{T}$. Write

$A_{1}=\left(\begin{array}[]{cccc}\lambda_{1}\\\ &&\ddots\\\
&&&\lambda_{k}\end{array}\right)\;\;\mbox{and}\;\;\;\;A_{2}=\left(\begin{array}[]{cccc}\lambda_{k+1}\\\
&&\ddots\\\ &&&\lambda_{m}\end{array}\right).$

Then, Equation (2.5) can be written as

$\displaystyle\frac{d}{dt}\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)(t)+\left(\begin{array}[]{cc}A_{1}\\\
&A_{2}\end{array}\right)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)(t)=\left(\begin{array}[]{c}I_{k\times k}\\\
0\end{array}\right)\alpha(t),$ (2.6)

together with the initial condition:

$(z_{1}(0),z_{2}(0))^{T}=\left(\left(\mathop{\langle}y_{0},\xi_{1}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{k}\mathop{\rangle}\right)^{T},\left(\mathop{\langle}y_{0},\xi_{k+1}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{m}\mathop{\rangle}\right)^{T}\right)^{T}$

This, along with the condition (1.2), indicates that $z_{2}(t)\neq 0$, for
each $t>0$ and each control $\alpha$ in ${\cal
V}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}$. Consequently, Problem $(\mathcal{P})$ has
no time optimal control.

Step 2. Suppose that $k<m$ and $y_{0}$ does not satisfy (1.2). Then, Problem
$(\mathcal{P})$ has optimal controls.

Let $\\{\bar{a}_{i}\\}_{i=1}^{k}$ be a finite sequence of positive numbers.
Since $y_{0}$ does not satisfy (1.2), it holds that $z_{2}(0)=0$. Thus, it
follows from (2.6) that $z_{2}(t)=0$ for all $t\geq 0$. Hence, Problem
$(\widetilde{\cal P})$ shares the same optimal time and optimal controls with
the following time optimal control problem:

$(\widetilde{{\cal P}}_{1}):\hskip
14.22636pt\displaystyle\inf\limits\left\\{t\geq
0\;\big{|}\;z_{1}(t;\alpha)=0\right\\},$

where the infimum is taken over all $\alpha$ from ${\cal
V}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}$, and $z_{1}(\cdot;\alpha)$ is the solution
to the equation:

$\dot{z}_{1}(t)+A_{1}z_{1}(t)=I_{k\times
k}\alpha(t),\;\;t\in\mathbb{R}^{+},\;z_{1}(0)=\left(\mathop{\langle}y_{0},\xi_{1}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{k}\mathop{\rangle}\right)^{T}.$

According to Lemma 2.1, Problem $(\widetilde{{\cal P}}_{1})$ has admissible
controls. Then, by the standard argument (see either Theorem 13 and the note
after it in Chapter III on Page 130 in [16] or Theorem 3.1 on Page 31 in [4]),
one can easily verify that Problem $(\widetilde{{\cal P}}_{1})$ has optimal
controls. Consequently, Problem $(\mathcal{P})$ has optimal controls.

Step 3. Suppose that $k\geq m$. Then, Problem $({\cal P})$ admits optimal
controls.

Let $\\{\bar{a}_{i}\\}_{i=1}^{k}$ be a finite sequence of positive numbers.
Since $B=\left(I_{m\times m},0_{m\times(k-m)}\right)$ in the case that $k\geq
m$, control variables $\alpha_{m+1}(\cdot),\cdots,\alpha_{k}(\cdot)$ play no
role in Equation (2.5) when $k>m$. Hence, in the case that $k\geq m$, the
effective controls in Problem $(\widetilde{P})$ have the form:
$\hat{\alpha}=(\alpha_{1}(\cdot),\cdots,\alpha_{m}(\cdot))^{T}$. Therefore,
Problem $(\widetilde{{\cal P}})$ shares the same optimal time and optimal
controls with the following time optimal control problem:

$(\widetilde{{\cal P}}_{2}):\hskip
14.22636pt\displaystyle\inf\limits\left\\{t\geq
0\;\big{|}\;z(t;\hat{\alpha})=0\right\\},$

where the infimum is taken over all
$\hat{\alpha}\triangleq(\alpha_{1},\cdots,\alpha_{m})^{T}$ from the control
constraint set:

$\displaystyle{\cal
V}_{\\{\bar{a}_{i}\\}_{i=1}^{m}}\triangleq\left\\{\alpha=(\alpha_{1},\cdots,\alpha_{m})^{T}\bigm{|}\;\mbox{each}\;\;\alpha_{i}\;\mbox{is
measurable
from}\;\mathbb{R}^{+}\;\;\mbox{to}\;\;[-\bar{a}_{i},\bar{a}_{i}]\right\\},$

and $z(\cdot;\hat{\alpha})$ is the solution of the following equation:

$\left\\{\begin{array}[]{l}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\dot{z}(t)+Az(t)=I_{m\times m}\hat{\alpha}(t),\quad
t\in\mathbb{R}^{+},\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
z(0)=z_{0}\triangleq\left(\mathop{\langle}y_{0},\xi_{1}\mathop{\rangle},\mathop{\langle}y_{0},\xi_{2}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{m}\mathop{\rangle}\right)^{T}.\end{array}\right.$
(2.7)

Then, by Lemma 2.1, using the same argument as that in Step 2, one can prove
that Problem $(\widetilde{{\cal P}})$ has optimal controls.

In summary, we complete the proof.

######  Remark 2.3.

From the proof of Theorem 2.2, one can easily verify the following: $(a)$
Suppose that $\omega=\Omega$. Let $y_{0}\notin\mathcal{S}_{m}$ and $k$ be
given. Then, Problem $(\mathcal{P})$ has optimal controls if and only if there
is a finite sequence of positive numbers $\\{\bar{b}_{i}\\}_{i=1}^{k}$ such
that the problem $(\mathcal{P})$, with $\\{\bar{b}_{i}\\}_{i=1}^{k}$, has
optimal controls. $(b)$ In the case that $\omega=\Omega$ and
$y_{0}\notin\mathcal{S}_{m}$, Problem $(\mathcal{P})$ has optimal controls,
provided either $k\geq m$ or $k<m$ and $y_{0}$ does not satisfy (1.2).

######  Theorem 2.4.

Let $\omega=\Omega$ and $y_{0}\notin{\cal S}_{m}$. Let
$\\{\bar{a}_{i}\\}_{i=1}^{k}$ be a finite sequence of positive numbers. For
each $i\in\\{1,\cdots,k\\}$, write

$T_{i}=\displaystyle\frac{1}{\lambda_{i}}\ln\left(1+\frac{\lambda_{i}}{\bar{a}_{i}}\left|\mathop{\langle}y_{0},\,\xi_{i}\mathop{\rangle}\right|\right).\qquad$
(2.8)

Then Problem $({\cal P})$, with $\\{\bar{a}_{i}\\}_{i=1}^{k}$, does not have
the bang-bang property, if either of the following conditions stands: $(i)$
$k\geq m$ and the numbers $T_{1},\cdots,T_{m}$ are not the same; $(ii)$ $k<m$,
$y_{0}$ does not satisfy (1.2) and the numbers $T_{1},\cdots,T_{k}$ are not
the same.

Proof. Simply write $(\mathcal{P})$ for the problem $({\cal P})$, with
$\\{\bar{a}_{i}\\}_{i=1}^{k}$. For each $i\in\\{1,\cdots k\\}$, define

$\displaystyle\widetilde{\alpha}_{i}(\cdot)=-\chi_{[0,T_{i}]}(\cdot)sgn\big{(}\mathop{\langle}y_{0},\xi_{i}\mathop{\rangle}\big{)}\bar{a}_{i}\triangleq\left\\{\begin{array}[]{cl}\vskip
3.0pt plus 1.0pt minus
1.0pt\cr\chi_{[0,T_{i}]}(\cdot)\bar{a}_{i},&\mbox{if}\;\mathop{\langle}y_{0},\xi_{i}\mathop{\rangle}<0,\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0,&\mbox{if}\;\mathop{\langle}y_{0},\xi_{i}\mathop{\rangle}=0,\\\ \vskip 3.0pt
plus 1.0pt minus
1.0pt\cr-\chi_{[0,T_{i}]}(\cdot)\bar{a}_{i},&\mbox{if}\;\mathop{\langle}y_{0},\xi_{i}\mathop{\rangle}>0.\end{array}\right.$
(2.12)

We first prove the following property $(\mathcal{H}_{1})$: When $k\geq m$,
$\widetilde{T}$ and $\widetilde{u}$ are the optimal time and an optimal
control to Problem $(\mathcal{P})$ respectively, where

$\widetilde{T}\triangleq\max\\{T_{1},T_{2},\cdots,T_{m}\\}\;\;\mbox{and}\;\;\widetilde{u}\triangleq\sum\limits_{i=1}^{m}\widetilde{\alpha}_{i}\xi_{i}.$

By the equivalence of Problems $(\mathcal{P})$ and $(\widetilde{{\cal
P}}_{2})$ (see Step 3 in the proof of Theorem 2.2), we need only to verify
that $\widetilde{T}$ and $\widetilde{\alpha}$ are the optimal time and an
optimal control to Problem $(\widetilde{{\cal P}}_{2})$ respectively, where
$\widetilde{\alpha}\triangleq(\widetilde{\alpha}_{1},\cdots,\widetilde{\alpha}_{m})^{T}$.

For this purpose, we observe from direct computation that for each
$i\in\\{1,\cdots,m\\}$, $T_{i}$ and $\widetilde{\alpha}_{i}(\cdot)$ are the
optimal time and the optimal control to the following time optimal control
problem:

$(P_{i}):\hskip 14.22636pt\displaystyle\inf\limits\left\\{t\geq
0\;\big{|}\;z_{i}(t;\alpha_{i})=0\right\\},$

where the infimum is taken over all $\alpha_{i}(\cdot)$ from the set of all
measurable functions from $R^{+}$ to $[-\bar{a}_{i},\bar{a}_{i}]$, and
$z_{i}(\cdot;\alpha_{i})$ solves the following equation:

$\dot{z}_{i}(t)+\lambda_{i}(t)z_{i}(t)=\alpha_{i}(t),\;\;z_{i}(0)=\mathop{\langle}y_{0},\xi_{i}\mathop{\rangle}.$

Clearly, $\widetilde{\alpha}\in{\cal V}_{\\{\bar{a}_{i}\\}_{i=1}^{m}}$ and
$(\left(z_{1}\left(\cdot;\widetilde{\alpha}_{1}\right),\cdots,z_{m}\left(\cdot;\widetilde{\alpha}_{m}\right)\right)^{T}$
is the solution $z\left(\cdot;\widetilde{\alpha}\right)$ to Equation (2.7)
with $\hat{\alpha}=\widetilde{\alpha}$. Since
$z_{i}\left(T_{i};\widetilde{\alpha}_{i}\right)=0$, it holds that

$\displaystyle
z_{i}\left(\widetilde{T};\widetilde{\alpha}_{i}\right)=0\;\;\mbox{ for
all}\;\;i\in\\{1,2,\cdots,m\\},\;\;\mbox{i.e.,}\;\;z\left(\widetilde{T};\widetilde{u}\right)=0.$
(2.13)

Hence, the optimal time to Problem $(\widetilde{{\cal P}}_{2})$ is not bigger
than $\widetilde{T}$. On the other hand, if
$\hat{\alpha}\triangleq(\hat{\alpha}_{1}\cdots,\hat{\alpha}_{m})^{T}\in{\cal
V}_{\\{\bar{a}_{i}\\}_{i=1}^{m}}$ and $\hat{T}>0$ are such that
$z\left(\hat{T};\hat{\alpha}\right)=0$, then it stands that

$z_{i}\left(\hat{T};\hat{\alpha}_{i}\right)=0\;\;\mbox{ for
all}\;\;i\in\\{1,\cdots,m\\}.$

By the optimality of $T_{i}$ to Problem $(P_{i})$, we see that $\hat{T}\geq
T_{i}$ for all $i\in\\{1,\cdots,m\\}$, from which, it follows that
$\hat{T}\geq\widetilde{T}$. Therefore, $\widetilde{T}$ is the optimal time to
Problem $(\widetilde{{\cal P}}_{2})$. Along with (2.13), this yields that
$\widetilde{\alpha}$ is an optimal control to this problem. Hence, the
property $(\mathcal{H}_{1})$ stands.

Since $y_{0}\notin{\cal S}_{m}$, it holds that $\widetilde{T}>0$. Because
$T_{1},\cdots,T_{m}$ are not the same, there is an
$i_{0}\in\\{1,2,\cdots,m\\}$ such that $T_{i_{0}}<\widetilde{T}$. Then, it
follows from (2.12) that $\widetilde{\alpha}_{i_{0}}(t)=0$ for all
$t\in(T_{i_{0}},\widetilde{T}]$. Thus, the optimal control $\widetilde{u}$
does not satisfy the bang-bang property.

Using the very similar argument to that in the proof of the property
$(\mathcal{H}_{1})$, one can easily show the following property
$(\mathcal{H}_{2})$: When $k<m$, $y_{0}$ does not satisfy (1.2), $\hat{T}$ and
$\hat{u}$ are the optimal time and an optimal control to Problem
$(\mathcal{P})$, where

$\hat{T}\triangleq\max\\{T_{1},T_{2},\cdots,T_{k}\\}\;\;\mbox{and}\;\;\hat{u}\triangleq\sum\limits_{i=1}^{k}\widetilde{\alpha}_{i}\xi_{i}.$

Then, by the property $(\mathcal{H}_{2})$, (2.12) and the assumptions that
$y_{0}\notin{\cal S}_{m}$ and the numbers $T_{1},\cdots,T_{k}$ are not the
same, one can easily show that the optimal control $\hat{u}$ does not satisfy
the bang-bang property. This completes the proof.

##  3 Studies of Problem $(\mathcal{P})$ where $\omega$ is a proper subset of
$\Omega$

######  Theorem 3.1.

Let $\Omega$ satisfy the condition $(D1)$. Suppose that $\omega$ holds the
condition $(D2)$. Then, for each $k\geq 1$, each $y_{0}\notin\mathcal{S}_{m}$
and each finite sequence of positive numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$,
Problem $(\mathcal{P})$ has optimal controls.

Proof. By the same way as that in Step 1 of the proof of Theorem 2.2, we
define the matrices $A$ and $B$, and the problem $(\widetilde{P})$. Write
$B_{ij}$ for the element in i-th row and j-th column of $B$, namely,
$B_{ij}=\mathop{\langle}\chi_{\omega}\xi_{i}\,,~{}\xi_{j}\mathop{\rangle}$.
Let $B_{1}=(B_{11},\cdots,B_{m1})^{T}$. We first claim that

$rank(B_{1},AB_{1},A^{2}B_{1},\cdots,A^{m-1}B_{1})=m.$ (3.1)

In fact, since

$A^{j}B_{1}=\left(\begin{array}[]{ccc}\lambda_{1}\\\ &\ddots\\\
&&\lambda_{m}\end{array}\right)^{j}\left(\begin{array}[]{c}B_{11}\\\ \cdots\\\
B_{m1}\end{array}\right)=\left(\begin{array}[]{c}\lambda^{j}_{1}B_{11}\\\
\cdots\\\ \lambda_{m}^{j}B_{m1}\end{array}\right),$

it holds that

$\Bigr{|}(B_{1},AB_{1},\cdots,A^{m-1}B_{1})\Bigl{|}=\left|\begin{array}[]{cccc}B_{11}&\lambda_{1}B_{11}&\cdots&\lambda_{1}^{m-1}B_{11}\\\
B_{21}&\lambda_{2}B_{21}&\cdots&\lambda_{2}^{m-1}B_{21}\\\
\cdots&\cdots&\cdots&\cdots\\\
B_{m1}&\lambda_{m}B_{m1}&\cdots&\lambda_{m}^{m-1}B_{m1}\end{array}\right|,$

which is a determinant of Vandermonde’ type and equals to
$\prod_{i=1}^{m}B_{i1}\prod_{k>l}(\lambda_{k}-\lambda_{l})$. Because of
conditions $(D1)$ and $(D2)$, this determinant is not zero, which implies
(3.1).

Now, according to Lemma 2.1, Problem $(\widetilde{\mathcal{P}})$ has
admissible controls. Then, by the standard argument (see either Theorem 13 and
the note after it in Chapter III on Page 130 in [16] or Theorem 3.1 on Page 31
in [4]), one can easily show that Problem $(\widetilde{{\cal P}})$ admits
optimal controls. This, along with the equivalence of Problems $(\mathcal{P})$
and $(\widetilde{{\cal P}})$, completes the proof.

######  Remark 3.2.

From the proof of the above theorem,it follows that Theorem 3.1 still stands
when the condition $(D2)$ is replaced by the following condition:

  * •

$(\widetilde{D}2)$
$\mathop{\langle}\chi_{\omega}\xi_{i}\,,~{}\xi_{1}\mathop{\rangle}\neq
0\quad{\rm for~{}all}\;\;i\in\\{1,2,\cdots,m\\}.$

Before studying the bang-bang property for Problem $(\mathcal{P})$ where
$\omega$ is a proper open subset of $\Omega$, we recall the general position
condition which plays an important role in the studies of the bang-bang
property for linear controlled ordinary differential equations. Let $\hat{A}$
and $\hat{B}$ be $m\times m$ and $m\times k$ matrices respectively. Let
$\hat{V}$ be a closed polyhedron in $\mathbb{R}^{k}$. We say that $\hat{V}$
satisfies the general position condition with respect to $(\hat{A},\hat{B})$,
if for each nonzero vector $v$, which is parallel to one of the edges of
$\hat{V}$, the vectors

$\hat{B}v,~{}\hat{A}\hat{B}v,~{}\cdots~{}\hat{A}^{m-1}\hat{B}v$

are linearly independent. Consider the following time optimal control problem:

$(\hat{P}):\hskip
56.9055pt\displaystyle\inf\limits\left\\{t:z(t;v,z_{0})=0\right\\},\hskip
99.58464pt$

where the infimum is taken over all measurable functions $v$ from
$\mathbb{R}^{+}$ to the polyhedron $\hat{V}$, and $z(\cdot;v,z_{0})$ is the
solution to the following equation:

$\dot{z}(t)+\hat{A}z(t)=\hat{B}v(t),\;\;t>0;\;\;z(0)=z_{0},$

with $z_{0}$ a non-zero vector in $\mathbb{R}^{m}$.

######  Lemma 3.3.

(see [16], [7]) Suppose that the closed polyhedron $\hat{V}$ satisfies the
general position condition with respect to $(\hat{A},\hat{B})$. Then any
optimal control $\bar{u}(t)$ to Problem $(\hat{P})$, if exists, takes values
on the vertices of $\hat{V}$ and has a finite number of switchings.

######  Theorem 3.4.

Let $\Omega$ satisfy the condition $(D1)$. Suppose that $\omega$ satisfies the
condition $(D2)$. Then, for each $k\geq 1$, each $y_{0}\notin\mathcal{S}_{m}$
and each finite sequence of positive numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$ ,
Problem $(\mathcal{P})$ holds the bang-bang property.

Proof. By the same way as that in Step 1 of the proof of Theorem 2.2, we
define the matrices $A$ and $B$, and the problem $(\widetilde{\mathcal{P}})$.
According to Lemma 3.3, Theorem 3.1 and the equivalence of Problems
$(\mathcal{P})$ and $(\widetilde{\mathcal{P}})$, it suffices to prove the
general condition of $U_{\\{\bar{a}_{i}\\}^{k}_{i=1}}$ with respect to
$(A,B)$. Clearly, the later is equivalent to the statement that for each
$j\in\\{1,\cdots,k\\}$, the vectors $Be_{j},ABe_{j},\cdots,A^{m-1}Be_{j}$ are
linearly independent, where $\\{e_{1},\cdots,e_{k}\\}$ is the standard basis
of $\mathbb{R}^{k}$.

Let $F_{j}=(Be_{j},ABe_{j},\cdots,A^{m-1}Be_{j})$. It is clear that

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr|F_{j}|=&|(Be_{j},ABe_{j},\cdots,A^{m-1}Be_{j})|\\\ \vskip 6.0pt plus
2.0pt minus
2.0pt\cr=&\left|\begin{array}[]{cccc}B_{1j}&\lambda_{1}B_{1j}&\cdots&\lambda_{1}^{m-1}B_{1j}\\\
B_{2j}&\lambda_{2}B_{2j}&\cdots&\lambda_{2}^{m-1}B_{2j}\\\
\cdots&\cdots&\cdots&\cdots\\\
B_{mj}&\lambda_{m}B_{mj}&\cdots&\lambda_{m}^{m-1}B_{mj}\end{array}\right|\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr=&\prod_{i=1}^{m}B_{ij}\prod_{k>l}(\lambda_{k}-\lambda_{l}).\end{array}$

This, together with the conditions (D1) and (D2), yields that $|F_{j}|\neq 0$
for each $j\in\\{1,\cdots,k\\}$. Hence, $U_{\\{\bar{a}_{i}\\}^{k}_{i=1}}$
satisfies the general position condition with respect to $(A,B)$. This
completes the proof.

##  4 Further studies on the conditions $(D1)$ and $(D2)$

In this section, we first give a remark and a theorem, which reveal
accordingly some properties for eigenvalues and eigenfunctions of $-\Delta$
with Dirichlet boundary condition. From the remark, it follows that there are
a lot of $\Omega$ satisfying the property $\bf(D1)$. From the theorem, it
follows that for any bounded domain $\Omega$ in $\mathbb{R}^{n}$, there are a
lot of $\omega\subset\Omega$ where the property $\bf(D2)$ holds. We end this
section with another remark which provides an open problem.

######  Remark 4.1.

It is presented in [12] (see also [18], [14]) that there are a lot of $\Omega$
of class $C^{3}$ satisfies the condition $(D1)$ in the following sense: Let
$\Omega$ be a bounded open set of class $C^{3}$ in $\mathbb{R}^{n}$. For each
$\varepsilon\in(0,1)$, an $\varepsilon-$neighborhood of $\Omega$ is defined to
be the image $(I+\psi)(\Omega)$, where $I$ is the identity map over
$\mathbb{R}^{n}$ and $\psi\in C^{3}(\mathbb{R}^{n};\mathbb{R}^{n})$, with the
$C^{3}-$norm less than $\varepsilon$. For each bounded open set
$\widetilde{\Omega}$ of class $C^{3}$ in $\mathbb{R}^{n}$, Write
$\Delta_{\widetilde{\Omega}}$ for the self-adjoint operator in
$L^{2}(\widetilde{\Omega})$ generated by the Laplacian on $\widetilde{\Omega}$
with the homogeneous Dirichlet boundary condition. Then, for each
$\varepsilon\in(0,1)$, there is an $\varepsilon-$neighborhood of
$\Omega^{\varepsilon}$ such that $-\Delta_{\Omega^{\varepsilon}}$ has only
simple eigenvalues.

Before presenting the theorem, we introduce the following notations: for each
$x\in\mathbb{R}^{n}$ and each $\rho>0$, $B_{\rho}(x)$ stands for the open ball
in $\mathbb{R}^{n}$, centered at $x$ and of radius $\rho$;
$\overline{B_{\rho}(x)}$ denotes the closure of the ball $B_{\rho}(x)$; for
each $\rho>0$,

$\Omega^{\rho}\triangleq\Bigr{\\{}x\in\Omega\setminus\overline{\omega}\bigm{|}dist\left(\partial{B_{\rho}(x)},\partial\Omega\right)>0,~{}dist\left(\partial{B_{\rho}(x)},\partial\omega\right)>0\Bigl{\\}},$

where $dist\left(E_{1},E_{2}\right)\triangleq\inf\limits_{x_{1}\in
E_{1},x_{2}\in E_{2}}\|x_{1}-x_{2}\|_{\mathbb{R}^{n}}$ for any subsets $E_{1}$
and $E_{2}$ in $\mathbb{R}^{n}$.

######  Theorem 4.2.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ and $\omega$ be an open
subset of $\Omega$ such that
$\Omega\setminus\overline{\omega}\neq\varnothing$. Then, for any
$\varepsilon>0$, there exists an $\varepsilon_{0}\in(0,\varepsilon)$ such that
$\Omega^{\varepsilon_{0}}\neq\varnothing$ and for almost every
$\widetilde{x}\in\Omega^{\varepsilon_{0}}$,

$\displaystyle\displaystyle\Bigl{\langle}\chi_{\omega\cup
B_{\varepsilon_{0}}(\widetilde{x})}\xi_{i}\,,~{}\xi_{j}\Bigr{\rangle}\neq
0\quad{\rm for~{}all~{}}i,j\in\mathbb{N}.$ (4.1)

Proof. We recall that each eigenfunction $\xi_{i}$ belongs to
$C^{\infty}(\Omega)$ (see Page 335 in [5]). Let

$\displaystyle\varphi(x,\tau)=\xi_{i}(x)\displaystyle
e^{\sqrt{\lambda_{i}}\tau},\qquad(x,\tau)\in\Omega\times\mathbb{R}.$

It is obvious that

$\triangle_{x}\varphi(x,\tau)+\partial^{2}_{\tau}\varphi(x,\tau)=0,\qquad(x,\tau)\in\Omega\times\mathbb{R}.$

By the property of harmonic functions (see Page 6 in [8]), the function
$\varphi(\cdot,\cdot)$ is real analytic over $\Omega\times\mathbb{R}$. Thus,
each eigenfunction $\xi_{i}$ is real analytic over $\Omega$. Write

$D=\left\\{(x,\rho)\in\Omega\times\mathbb{R}^{+}\bigm{|}\overline{B_{\rho}(x)}\subset\Omega\right\\}.$

Then, for each pair $(i,j)$, we define a function $F_{i,j}(\cdot.\cdot)$ from
$D$ to $\mathbb{R}$ by setting:

$F_{i,j}(x,\rho)=\displaystyle\int_{B_{1}(0)}\xi_{i}(x+\rho\eta)\xi_{j}(x+\rho\eta)d\eta,\;\;(x,\rho)\in
D.$ (4.2)

Clearly, it is well defined. The rest of the proof will be carried out by the
following three steps:

Step 1. Suppose that $f$ is a real analytic function over $\Omega$. Define the
function $F:D\mapsto\mathbb{R}$ by

$F(x,\rho)=\displaystyle\int_{B_{1}(0)}f(x+\rho\eta)d\eta,\;\;(x,\rho)\in D.$
(4.3)

Then $F$ is real analytic over $D$.

We need only to explain that $F$ is real analytic in a small neighborhood of
$(x_{0},\rho_{0})$ for any point $(x_{0},\rho_{0})\in D$. First, there is a
neighborhood $U$ of $(x_{0},\rho_{0})$ in $\mathbb{R}^{n}\times\mathbb{R}^{+}$
such that $\overline{B_{\rho}(x)}\subset\Omega$ for any $(x,\rho)\in U$.
Hence, the function $f(x+\rho\eta)$ is real analytic in $(x,\rho,\eta)$ over
$U\times\\-{B_{1}(0)}$. Extend $f$ to a complex-valued function in
$(z,w,\eta)$ over a small neighborhood $U_{c}\times{B_{1}(0)}$ of
$U\times{B_{1}(0)}$ in ${\mathbb{C}}^{n}\times{\mathbb{C}}\times\\-{B_{1}(0)}$
by making use of the power series expansion. We then get $f_{c}(z+w\eta)$,
which is real analytic over $(z,w,\eta)\in U_{c}\times\\-{B_{1}(0)}$ and
holomorphic in $(z,w)\in U_{c}$ for each fixed $\eta\in\overline{{B}_{1}(0)}$.
Clearly, it holds that

$f_{c}(x+\rho\eta)=f(x+\rho\eta)\qquad{\rm for~{}all~{}}(x,\rho,\eta)\in
U\times B_{1}(0).$

Now we define a function $F_{c}:U_{c}\mapsto\mathbb{C}$ by setting:

$F_{c}(z,w)=\displaystyle\int_{B_{1}(0)}f(z+w\eta)d\eta,\;\;(z,w)\in U_{c},$

and define the operator $\bar{\partial}$ in the standard way:

$\displaystyle\bar{\partial}u(z,w)=\sum^{n}_{j=1}\frac{\partial
u}{\partial\bar{z}_{j}}d\bar{z}_{j}+\frac{\partial
u}{\partial\bar{w}}d\bar{w},$

where

$\displaystyle\frac{\partial}{\partial\bar{z}_{j}}=\frac{1}{2}\left(\frac{\partial}{\partial(Re(z_{j}))}+\sqrt{-1}\frac{\partial}{\partial(Im(z_{j}))}\right),\qquad\frac{\partial}{\partial\bar{w}}=\frac{1}{2}\left(\frac{\partial}{\partial(Re(w))}+\sqrt{-1}\frac{\partial}{\partial(Im(w))}\right)$

are the standard Cauchy-Riemann operators (see [9]). It follows from the
holomorphic property of $f_{c}$ in $(z,w)$ that

$\bar{\partial}F_{c}(z,w)=\displaystyle\int_{B_{1}(0)}\bar{\partial}f_{c}(z+w\eta)d\eta=\int_{B_{1}(0)}0~{}d\eta=0.$

Hence, $F_{c}$ is holomorphic in $U_{c}$. In particular, the function
$F_{c}(\cdot,\cdot)\Bigl{|}_{U_{c}\bigcap(\mathbb{R}^{n}\times\mathbb{R})}$ is
real analytic, i.e. $F(\cdot,\cdot)$ is analytic in $U$. Thus,
$F(\cdot,\cdot)$ is real analytic over $D$. Consequently, for each pair
$(i,j)$, the function $F_{i,j}(\cdot,\cdot)$ is real analytic over $D$.

Step 2. For each pair $(i,j)$, $F_{i,j}(\cdot,\cdot)$ is not identically a
constant over $D$.

By the unique continuation property of the eigenfunctions (see [10]), we see
that for each $i\in\\{1,2,\cdots\\}$,

$\xi_{i}(x)\neq 0\quad\mbox{for almost every}\quad x\in\Omega.$

Thus, it holds that for each pair $(i,j)$,

$(\xi_{i}\xi_{j})(x)\neq 0\quad\mbox{for almost every}\quad x\in\Omega.$

Since the function $(\xi_{i}\xi_{j})(\cdot)$ is continuous in $\Omega$ and
$\Omega\setminus\overline{\omega}\neq\varnothing$, there is an
$\hat{x}\in\Omega\setminus\overline{\omega}$ such that
$(\xi_{i}\xi_{j})(\hat{x})\neq 0$. Hence, when $\delta>0$ is small enough, the
function $\bigr{(}\xi_{i}\xi_{j}\bigl{)}(\cdot)$ is either positive or
negative over $B_{\delta}(\hat{x})$, and
$\overline{B_{\delta}(\hat{x})}\subset\Omega$. Now, it follows from the
definition of the function $F_{i,j}(\cdot,\cdot)$ that

$F_{i,j}(\hat{x},\delta_{1})\neq F_{i,j}(\hat{x},\delta_{2}),\;\;{\rm
when}~{}\delta_{1}\;\mbox{and}\;\delta_{2}\;\mbox{are different numbers
in}\;\;(0,\delta).$

Since $(\hat{x},\delta_{1})$ and $(\hat{x},\delta_{2})$ belong to $D$,
$F_{i,j}$ is not identically zero over $D$ for each pair $(i,j)$.

Step 3. To prove (4.1).

Since each $F_{i,j}$ is real analytic and is not identically a constant over
$D$, the set

$W_{i,j}\triangleq\left\\{(x,\rho)\in
D\bigm{|}F_{i,j}(x,\rho)+\mathop{\langle}\chi_{\omega}\xi_{i},\xi_{j}\mathop{\rangle}=0\right\\}$

is a real analytic subvariety with dimension at most $n$. Thus, the
$\mathbb{R}^{n+1}-$Lebesgue measure of the set

$W\triangleq\bigcup_{i,j}W_{i,j}$

is zero. Write $W^{\rho}=\left\\{x\in\Omega\bigm{|}(x,\rho)\in W\right\\}$.
Denote by $m(W^{\rho})$ the $\mathbb{R}^{n}-$Lebesgue measure of $W^{\rho}$.
According to Fubini’s Theorem,

$0=\displaystyle\int_{\Omega\times(0,\infty)}\chi_{W}(x,\rho)dxd\rho=\int^{\infty}_{0}\int_{\Omega}\chi_{W^{\rho}}(x)dxd\rho=\int^{\infty}_{0}m(W^{\rho})d\rho.$

Thus, there is a subset $E\subset(0,\infty)$ of zero measure such that
$m(W^{\rho})=0$ for each $\rho\in(0,\infty)\setminus E$. On the other hand,
since $\Omega\setminus\overline{\omega}\neq\varnothing$, there is
$\bar{\rho}>0$ such that $\Omega^{\rho}\neq\varnothing$ for all
$\rho\in(0,\bar{\rho}]$.

Now, for each $\varepsilon>0$, we arbitrarily take an $\varepsilon_{0}$ from
the set $(0,\min\\{\bar{\rho},\varepsilon\\})\setminus E$. Then, it stands
that $m\left(W^{\varepsilon_{0}}\right)=0$ and
$\Omega^{\varepsilon_{0}}\neq\varnothing$. Hence,

$\displaystyle m\left(\Omega^{\varepsilon_{0}}\setminus
W^{\varepsilon_{0}}\right)=m\left(\Omega^{\varepsilon_{0}}\right).$ (4.4)

Clearly, the statement that $x\in\Omega^{\varepsilon_{0}}\setminus
W^{\varepsilon_{0}}$ is equivalent to the statement that

$x\in\Omega^{\varepsilon_{0}}\;\;\mbox{and}\;\;F_{i,j}(x,\varepsilon_{0})+\mathop{\langle}\chi_{\omega}\xi_{i},\xi_{j}\mathop{\rangle}\neq
0\;\;\mbox{for all}\;\;i,j\in\mathbb{N}.$

This, together with (4.4), yields that

$\displaystyle
F_{i,j}(\widetilde{x},\varepsilon_{0})+\mathop{\langle}\chi_{\omega}\xi_{i},\xi_{j}\mathop{\rangle}\neq
0\;\;\mbox{for all}\;\;i,j\in\mathbb{N}\;\;\mbox{and for almost
every}\;\;\widetilde{x}\in\Omega^{\varepsilon_{0}}.$ (4.5)

Finally, by the definition of $\Omega^{\varepsilon_{0}}$, we see that

$B_{\varepsilon_{0}}(x)\subset\Omega\;\;\mbox{and}\;\;B_{\varepsilon_{0}}(x)\bigcap\omega=\varnothing\;\;\mbox{for
all}\;\;x\in\Omega^{\varepsilon_{0}}.$

Along with (4.5), these indicate (4.1). This completes the proof.

######  Remark 4.3.

Let $\\{a_{i}\\}_{i=1}^{\infty}\in
l^{2}_{+}\triangleq\left\\{\\{b_{i}\\}_{i=1}^{\infty}\in
l^{2}\;\big{|}\;b_{i}>0\;\;\mbox{for all}\;\;i\right\\}$. Consider the
following time optimal control problem $(P)$:
$\displaystyle\inf\limits\left\\{t:y(t;u,y_{0})=0\right\\}$, where the infimum
is taken over all $u$ from the set:

$U_{ad}=\left\\{u=\sum_{i=1}^{\infty}u_{i}(t)\xi_{i}\biggm{|}\;\mbox{each}\;\;u_{i}(\cdot)\;\;\mbox{is
measurable from}\;\;\mathbb{R}^{+}\;\;\mbox{to}\;\;[-a_{i},a_{i}]\right\\},$

and $y(\cdot;u,y_{0})$ is the solution to Equation (1.1). The set $U_{ad}$ is
called a control constraint set of the rectangular type. We say Problem $(P)$
has the bang-bang property if any optimal control
$u^{*}=\sum_{i=1}^{\infty}u^{*}_{i}(t)\xi_{i}$ satisfies that for each $i$,
$u^{*}_{i}(t)=a_{i}$ for a.e. $t\in(0,t^{*})$, where $t^{*}$ is the optimal
time.

It is not clear to us what conditions are needed to obtain the bang-bang
property for Problem $(P)$. With regard to this question, we would like to
mention the following: $(i)$ It is necessary to impose certain conditions on
$\\{a_{i}\\}_{i=1}^{\infty}\in l^{2}_{+}$ to ensure the existence of optimal
controls for Problem $(P)$ (see [11]); $(ii)$ When $U_{ad}$ is replaced by the
following control constraint sets of the ball type:

$\widetilde{U}_{ad}\triangleq\left\\{u(\cdot)\in
L^{\infty}(\mathbb{R}^{+};L^{2}(\Omega))\biggm{|}u(t)\in\widetilde{B}(0,r)\right\\},$

where $\widetilde{B}(0,r)$ is the ball in $L^{2}(\Omega)$, centered at the
origin and of radius $r>0$, the bang-bang property for the corresponding time
optimal control problem $(P)$ has been studied (see [6], [13], [19] and [15]).

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