New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints

Abstract

We study first-order necessary optimality conditions and finite element error estimates for a class of distributed parabolic optimal control problems with pointwise state constraints. It is demonstrated that, if the bound in the state constraint and the differential operator in the governing PDE fulfill a certain compatibility assumption, then locally optimal controls satisfy a stationarity system that allows to significantly improve known regularity results for adjoint states and Lagrange multipliers in the parabolic setting. In contrast to classical approaches to first-order necessary optimality conditions for state-constrained problems, the main arguments of our analysis require neither a Slater point, nor uniform control constraints, nor differentiability of the objective function, nor a restriction of the spatial dimension. As an application of the established improved regularity properties, we derive new finite element error estimates for the dG(0)-cG(1)-discretization of a purely state-constrained linear-quadratic optimal control problem governed by the heat equation. The paper concludes with numerical experiments that confirm our theoretical findings.

On the nonuniqueness and instability of solutions of tracking-type optimal control problems

“…As H 1 (0, T ; L 2 (Ω)) embeds continuously into C([0, T ]; L 2 (Ω)) by [31,Theorem 10.9], the above implies that (9) possesses a well-defined, weak-to-weak continuous control-to-state (or, in this context, more precisely control-to-observation) operator cannot be true a.e. in Ω for all α ∈ R. This shows that the map S is indeed nonaffine in the situation of ( 9) and (10). In summary, we may now again conclude that ( 9) satisfies all of the conditions in Assumption 1.1 (with p := 2, Y := L 2 (Ω), U := L 2 (0, T ; L 2 (D)), and appropriately rescaled norms).…”

“…Lastly, we would like to mention that studying the (non)uniqueness of solutions of P(y d ,u d ) becomes much more involved if Y and U are not assumed to be uniformly smooth and uniformly convex and if the exponent p is also allowed to take the value one. (Such cases occur, for instance, in the context of bang-bang and L 1 -tracking-type optimal control problems, see [7] and [10,Example 3.11].) On the one hand, in spaces that are not uniformly smooth and uniformly convex, solutions of problems of the form P(y d ,u d ) can be nonunique even when S is the identity map.…”

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

“…We remark that, in a less general format, the above result has already been used in [15], proof of Lemma A.1 and [16], proof of Theorem 2.2. By applying Lemma 4.1 to the PDE (2.3), it is straightforward to check that the directional derivative S (u; •) indeed satisfies the identity (1.2).…”

“…An extension to the parabolic setting, cf. [38], is also possible by invoking the results in the appendix of [15]. We restrict our attention to the setting in (P) to avoid obscuring the basic ideas of our analysis with unnecessary technicalities.…”

Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities

“…Suppose now that u 1 , u 2 ∈ Ũ and p 1 , p 2 ∈ P satisfying u 1 ≤ u 2 and p 1 ≤ p 2 are given and define y 1 := S(p 1 , u 1 ) and y 2 := S(p 2 , u 2 ). Then it follows from [16, Lemma A.1] and [35] that y 1 − max(0, y 1 − y 2 ), y 2 + max(0, y 1 − y 2 ) ∈ L 2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) holds and we may use (6.10) and the formulas in [16,Lemma A.1] to obtain (analogously to the elliptic case in Lemma 6.3(ii)) (6.12)…”