Optimal Control of Nonsmooth, Semilinear Parabolic Equations

Abstract

This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable, but not Gâteaux-differentiable. Two types of necessary optimality conditions are derived, the first one by means of regularization, the second one by using the directional differentiability of the control-to-state mapping. The paper ends with the application of the general results to a semilinear heat equation involving the max-function.

Robust adaptive boundary control of semilinear PDE systems using a dyadic controller

“…Theorem 3. Consider the closed-loop system (3), (7), (8), (10), and (11). Then, the state w is bounded in the sense of W for all times if Assumption 5 is satisfied.…”

“…26 This contrasts with the existing approaches to solving optimal control problems for semilinear systems. 10,24 On the other hand, it allows us to guarantee robustness rigorously using the small gain theorem as shown in this paper.…”

Optimal Control of a Viscous Two‐Field Gradient Damage Model

“…The literature here is rather scarce. We refer to [3,14,22] (elliptic VIs) and to [19] (non-smooth parabolic equations). Until recently, it was an open question whether such a system can be derived in the absence of 'ample controls', see also [22,Section 4].…”

“…For the optimal control of the parabolic obstacle problem, a strong stationarity system can be www.gamm-mitteilungen.org found in [21], but no rigorous proof is given there. Recently, an optimality system of strong stationary type was derived in [19] for an optimal control problem governed by a non-smooth parabolic PDE. This was possible due to the presence of so-called 'ample controls', which are necessary for deriving strong stationarity in most existing contributions, see e.g.…”

On an optimal control problem for non‐smooth wave phenomena

“…Furthermore, µ > 0 and η > 0 denote the control cost terms. Our goal is to prove an existence result and optimality conditions for (P), which require a substantial extension of the established techniques in [1][2][3][4].…”