We investigate the value function V:R+×Rn→R+∪{+∞} of the infinite horizon problem in optimal control for a general—not necessarily discounted—running cost and provide sufficient conditions for its ...lower semicontinuity, continuity, and local Lipschitz regularity. Then we use the continuity of V(t,⋅) to prove a relaxation theorem and to write the first order necessary optimality conditions in the form of a, possibly abnormal, maximum principle whose transversality condition uses limiting/horizontal supergradients of V(0,⋅) at the initial point. When V(0,⋅) is merely lower semicontinuous, then for a dense subset of initial conditions we obtain a normal maximum principle augmented by sensitivity relations involving the Fréchet subdifferentials of V(t,⋅). Finally, when V is locally Lipschitz, we prove a normal maximum principle together with sensitivity relations involving generalized gradients of V for arbitrary initial conditions. Such relations simplify drastically the investigation of the limiting behavior at infinity of the adjoint state.
Given $\alpha \in 0,2)$ and $f \in L^2 ((0,T)\times(0,1))$, we derive new Carleman estimates for the degenerate parabolic problem $w_t + (x^\alpha w_x) _x =f$, where $(t,x) \in (0,T) \times (0,1)$, ...associated to the boundary conditions $w(t,1)=0$ and $w(t,0)=0$ if $0 \leq \alpha <1$ or $(x^\alpha w_x)(t,0)=0$ if $1\leq \alpha <2$. The proof is based on the choice of suitable weighted functions and Hardy-type inequalities. As a consequence, for all $0 \leq \alpha <2$ and $\omega\subset\subset(0,1)$, we deduce null controllability results for the degenerate one-dimensional heat equation $u_t - (x^\alpha u_x)_x = h \chi _\omega$ with the same boundary conditions as above.
We consider the typical one-dimensional strongly degenerate parabolic operator
Pu
=
u
t
− (
x
α
u
x
)
x
with 0 <
x
<
ℓ
and
α
∈ (0, 2), controlled either by a boundary control acting at
x
=
ℓ
, or by ...a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter
α
. We prove that the control cost blows up with an explicit exponential rate, as
e
C
/((2−
α
)
2
T
)
, when
α
→ 2
−
and/or
T
→ 0
+
. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions
J
ν
of large order
ν
(obtained by ordinary differential equations techniques).
In a separable Banach space E, we study the invariance of a closed set K under the action of the evolution equation associated with a maximal dissipative linear operator A perturbed by a ...quasi-dissipative continuous term B. Using the distance to the closed set, we give a general necessary and sufficient condition for the invariance of K. Then, we apply our result to several examples of partial differential equations in Banach and Hilbert spaces.
It is well-known that the value function V of a Bolza optimal control problem fails to be everywhere differentiable. In this paper, however, we show that, if V is proximally subdifferentiable at ...(t,x), then it is smooth on a neighborhood of (t,x). Our result yields that V stays smooth on a neighborhood of any optimal trajectory starting at a point where the proximal subdifferential is nonempty. This leads to sufficient conditions for the regularity of optimal trajectories and optimal controls.
We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domainΩ=(−1,1)×T×T taking as observation regions slices of the form ω=(a,b)×T×T or tubes ...ω=(a,b)×ωy×T, with −1<a<b<1. We prove that observability fails for an arbitrary time T>0 but both observability and Lipschitz stability hold true after a positive minimal time, which depends on the distance between ω and the boundary of Ω:Tmin⩾18min{(1+a)2,(1−b)2}. Our proof follows a mixed strategy which combines the approach by Lebeau and Robbiano, which relies on Fourier decomposition, with Carleman inequalities for the heat equations that are solved by the Fourier modes. We extend the analysis to the unbounded domain (−1,1)×T×R.
In this paper we study controllability properties of semilinear degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate ...results of ‘regional null controllability’, showing that we can drive the solution to rest at time
T on a subset of the space domain, contained in the set where the equation is nondegenerate.
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