Optimal Control Problems with Final Observation Governed by Explosive Parabolic Equations

Abstract

Abstract. We study optimal controls problems with final observation. The governing parabolic equations or systems involve superlinear nonlinearities and their solutions may blow up in finite time. Our proof of the existence, regularity and optimality conditions for an optimal pair is based on uniform a priori estimates for the approximating solutions. Our conditions on the growth of the nonlinearity are essentially optimal. In particular, we also solve a longstanding open problem of J.L. Lions concerning singular systems.

Optimal Control of the Thermistor Problem in Three Spatial Dimensions, Part 1: Existence of Optimal Solutions

“…(iii) Let Assumption 3.4 hold true for some q ∈ (3,4). Then the operator A(ζ) satisfies maximal parabolic regularity on for the assertions to hold with s = r. We will use this special case frequently in the course of the remaining part of this section.…”

“…Both terms prevent a blow-up of the temperature and its gradient and allow to restrict the discussion of the optimization problem to control functions that admit a global-intime solution of the state system. This approach is inspired by [4], where a similar technique was used to establish the existence of optimal controls. Let us put our work into perspective.…”

“…In the proof of Theorem 4.16, boundedness of minimizing sequence (u n ) in the control space U was essential and followed from the radial unboundedness of the objective functional as seen in (4.12). Alternatively, one could also assume that the upper bound u max in the control constraints satisfies u max ∈ L p (J; L p (Γ N )) with p > 4 3 q − 2. In this case, an objective functional of the form…”

A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems

“…These estimates are of independent interest since such bounds are known to have applications to blow-up [24,40], to the existence of periodic solutions [41], and to control problems [6]. Unfortunately we need to impose a technical restriction on the growth of g. In order to give an idea of our results assume for simplicity that g(u) = |u| p−1 u + g 1 (u) where g 1 ∈ C 1 (R, R) has bounded derivative and g 1 (0) = g 1 (0) = 0, and suppose that h ≡ 0.…”

Optimal control problems for some ordinary differential equations with behavior of blowup or quenching

“…It is one purpose of this paper. Optimal control problems for some partial differential equations with the property of blowup have been studied in few papers (see, for instance, [1], [17]). However, the intention of these papers was to find an optimal control with the minimal energy 810 PING LIN AND WEIHAN WANG from all the controls whose corresponding solutions exist in the closed interval where the cost functional is defined.…”