2.4 The case of a terminal condition on the state
In this part, we will conclude another generalization of the approach taken inSection 2.2. More specifically, in addition to the initial condition, we will allow for a terminal condition on the state in Problem 2.23. In order to not hide the main ideas behind technical details, we will assume bounded control and feedback operators and discuss the case of an unbounded but admissible control operator in Remark 2.53. Intuitively, it is clear that in order to satisfy this constraint, the set of prescribed terminal values needs to be reachable in the sense that there is a control that steers the initial state to the specified terminal state. This concept is called controllability and we will briefly introduce it in the following subsection.
2.4.1 Observability and controllability
For an overview of controllability and observability of finite dimensional systems the interested reader is referred to the overview given in [83, Chapters 2-4] and [159, Chapter 2] or [130].
There exist many characterizations for controllability and observability, e.g., the Kalman rank condition, the Hautus test or observability estimates. A very important property is the duality of observability and controllability, similarly to the duality of detectability and stabilizability, cf. Assumption 2.32orAssumption 2.45. Some concepts and properties of a finite dimensional setting carry over to an infinite dimensional setting; in particular, observability estimates or the duality mentioned above remain a very useful tool in the study of controllability and observability in infinite dimensions. However, as can be expected, there are some major differences in the infinite dimensional setting. First, there are several different concepts of controllability resp.
observability, namely approximate and exact controllability resp. observability and additionally the notion of null controllability. For linear, time reversible systems, null controllability and exact controllability are equivalent, cf. [159, Remark 3.1 b)] or [139, Remark 6.1.2]. Moreover, other than in finite dimensions, controllability resp. observability at some timetc>0 does not imply controllability resp. observability for all timest >0. The reader is referred to [34,35,139,159]
for an in-depth introduction to controllability and observability of infinite dimensional systems.
For anyτ ∈[0, T], let us recall the input mapφτ:L2(0, T;U)→X with φτu:=
Z τ 0
T(τ −s)Bu(s)ds,
as defined in Definition 2.39 i). In the following, we will assume (T(t))t≥0 to always be the semigroup generated by A:D(A)⊂X→X.
Definition 2.50. (Exact and approximate controllability) We call(A, B) exactly controllable in time tc > 0 if ranφtc = X. Similarly, we call (A, B) approximately controllable in time tc if ranφtc =X.
It is clear that exact and approximate controllability coincide in finite dimensions. More-over, it is obvious that exact controllability implies exponential stabilizability as defined in Assumption 2.32. An important characterization of controllability is the following observability inequality, which was proven first in the seminal paper [97] by the Hilbert Uniqueness Method.
Theorem 2.51 ([35, Theorem 4.1.7]). (A, B) is exactly controllable in time tc >0 if and only if there is αtc >0 such that
Z tc
0
kB∗T∗(s)x0k2U ds≥αtckx0k2 ∀x0 ∈X.
Using substitution in the previous estimate we immediately obtain that Z T
T−tc
kB∗T∗(T−s)λTk2Uds≥αtckλTk2 ∀λT ∈X. (2.52) 2.4.2 Scaling results and T-independent bounds
In this section, we discuss an extension of our result to optimal control problems with a condition on the terminal state, i.e., adding a terminal conditionx(T) =xT ∈X inProblem 2.23. In the finite dimensional case, turnpike results for linear and nonlinear initial and terminal conditions were proven in [136]. In the Hilbert space setting, this problem was discussed in [135, Section 2.6], where, however, the lack of invertibility of the Lyapunov operator prohibited the derivation of a result for a terminal condition. To ensure existence of an optimal solution for arbitrary initial and terminal data, we assume (A, B) to be exactly controllable in time tc where 0 < tc ≤ T, i.e., for any initial datum x0 ∈X and terminal state xT ∈X, we can find a control that drives the state fromx0 toxT in any time T ≥tc. This assumption excludes parabolic equations with control that does not act on the whole domain. The assumption is, however, fulfilled by many hyperbolic systems, cf.Section 2.5. For a discussion of controllability issues for PDEs, the reader is referred to the overview article [159].
Another crucial point in deriving Pontryagin Maximum Principles for problems including both initial and terminal conditions on the state in infinite dimensions is a codimensionality condition of the reachable set in X, cf. [95, Chapter 4]. This assumption is automatically satisfied if one assumes exact controllability, as the reachable set is the whole space X. For bounded control and observation operators, the optimal solutions satisfy the dynamics (2.4) with x(0) = x0 and x(T) = xT, where no terminal condition on the adjoint is imposed [95, Theorem 1.6]. Again, by eliminating the control, we obtain the extremal equations
We observe that in contrast to the initial condition on the state and the terminal condition on the adjoint equation in (2.27), the system (2.53) is subject to an initial and terminal condition on the state and no condition on the adjoint. As a consequence, the estimate presented in Lemma 2.36contains the unknown valueλ(T). In order to bound this unknown quantity by the right hand side of (2.53), we will utilize the following observability estimate.
2.4. THE CASE OF A TERMINAL CONDITION ON THE STATE
and (A, B) be exactly controllable in time tc. Then there isc >0 independent of T, such that kλ(T)k2≤c
Z T T−tc
kB∗λ(s)k2U+kCx(s)k2Y +kl1(s)k2ds.
Proof. The proof of this estimate is inspired by [113, Proof of Remark 2.1], where the finite dimensional case is considered. We decomposeλ=λ1+λ2, where
−λ01=A∗λ1, λ1(T) =λ(T),
−λ02=A∗λ2−C∗Cx+l1, λ2(T) = 0 , and apply the observability estimate (2.52) to λ1(t) =T∗(T −t)λ(T), which yields
αtckλ(T)k2 ≤
Similar to [113, Proof of Remark 2.1], it is important that we use integrals over time periods of length tc, which yields the constants in the proof of Proposition 2.52, in particular αtc and c(tc), independent ofT.
Having derived the desired estimate for the terminal state on the adjoint, we briefly comment on a possible extension to the unbounded case.
Remark 2.53. Controllability or observability with an unbounded but admissible control resp.
observation operator is discussed in, e.g., [34, Chapter 2] or [139, Chapter 6]. In that context, (2.52) is required for all x0 ∈ D(A∗), similarly to the admissibility inequality (2.46), cf. [34, Theorem 2.4.2] or [139, Definition 6.1.1]. In the above analysis, we observe that in the last estimate of the proof of Proposition 2.52, only an admissibility-like estimate of B is needed.
Now, similarly toTheorem 2.38and Theorem 2.47, we obtain a T-independent bound under an exact controllability assumption.
Theorem 2.54. Let(A, B)be exactly controllable in time0< tc≤T and(A, C)be exponentially
Proof. We proceed analogously to Section 2.2.3. In order to estimate the unknown λ(T) that appears on the right hand side of (2.42) and (2.44), we use the estimate obtained in Propo-sition 2.52. Thus, the statement (2.34) of Lemma 2.36 holds with an upper bound depending only on kl1k22+kl2k21∨2+kx0k2
. UsingLemma 2.37 and the fact thatx(T) =xT is a datum, a straightforward adaption of the proof of Theorem 2.38, wherexT plays the role of λT, yields the result.
Completely analogously to Theorem 2.27, we can derive an estimate on the propagation of perturbations.
Theorem 2.55. Let(A, B)be exactly controllable in time0< tc≤T and(A, C)be exponentially detectable. Let(ε1, ε2)∈L2(0, T;X)2 and (ε0, εT)∈X2. Assume (δx, δλ)∈C(0, T;X)2 solve
Then there is a scaling factor µ >0 satisfying
µ < 1
fM−1
L((L2(0,T;X)×X)2,L2(0,T;X)2)
and a constant c≥0, both independent of T, such that, defining ρ:=
Moreover, the corresponding counterpart of the turnpike result Theorem 2.30 in case of terminal conditions reads: