Strong stability of linear parabolic time-optimal control problems

Abstract

Sufficient conditions for strong stability of a class of linear time-optimal control problems with general convex terminal set are derived. Strong stability in turn guarantees qualified optimality conditions. The theory is based on a characterization of weak invariance of the target set under the controlled equation. An appropriate strengthening of the resulting Hamiltonian condition ensures strong stability and yields a priori bounds on the size of multipliers, independent of, e.g., the initial point or the running cost. In particular, the results are applied to the control of the heat equation into an L2-ball around a desired state.

Time-optimality by distance-optimality for parabolic control systems

“…Bu n − Ay n , p n dt with θ n → 0, y n = S(ν + θ n , u n ), and p n the associated adjoint state with terminal value y n (1) − y d . Convergence of ν n → ν, weak convergence of u n u, and compactness of (ν, u) → S(ν, u) from R + × L s (I × ω) to C([0, 1]; H), see [2,Proposition A.19], yields p n → p in W (0, 1). Hence In summary, this proves…”

A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Time-Optimal Control Problems

“…The target set U = { u ∈ L 2 (Ω) : G(u) ≤ 0 } is called weakly invariant under the state equation if, for any u 0 satisfying G(u 0 ) ≤ 0, there is a admissible control q(t) ∈ Q ad such that the corresponding trajectory with initial value u 0 satisfies G(u(t)) ≤ 0 for all times; cf. [4,Section 4] and the references therein. Since the formulation of (P ) only requires the state to be inside the target set at the final time T (but not at later times), it seems to be desirable to require the target set to be weakly invariant, since this guarantees that G(u(t)) ≤ 0 can be maintained for t > T .…”

“…Since the formulation of (P ) only requires the state to be inside the target set at the final time T (but not at later times), it seems to be desirable to require the target set to be weakly invariant, since this guarantees that G(u(t)) ≤ 0 can be maintained for t > T . However, this requirement already implies that the metric projection P U to U in L 2 (Ω) is stable in H 1 0 (Ω); see [4,Lemma 3.5]. This further leads to the requirement G (P U (u)) * = P U (u) − u d ∈ H 1 0 (Ω) for all u ∈ H 1 0 (Ω), which implies the assumption on u d .…”

Error Estimates for Space-Time Discretization of Parabolic Time-Optimal Control Problems with Bang-Bang Controls

“…Note first that the linearized Slater condition allows for exact penalization of (P ); see [6, Theorem 2.87, Proposition 3.111]. The optimality conditions now follow as in the proof of [4,Theorem 4.12]. The condition (3.6) is equivalent to ∂ ν L(ν,q,μ) = 0 and (3.7) arises from (3.5) for δν = 0.…”

“…In [22] an error estimate for the optimal times is proved that does not require uniqueness of the solution. However, in the particular case considered there the linearized Slater condition holds uniformly for the discrete problem, and this would also suffice for our argument; we also refer to [2,Section 5.6] for a generalization of [22] to fully discrete problems. For the case of a distributed control with the variational control discretization we can improve the result of [18] (see also [38] for a semilinear state equation) and obtain an optimal rate O(k + h 2 ).…”