Time Optimal Controls of Semilinear Heat Equation with Switching Control

Abstract

This paper is concerned with the bang-bang property of time optimal controls, governed by a semilinear heat equation in a bounded domain with switching controls acting locally into two open subsets. The proofs rely on an observability estimate from a positive measurable set in time for the linear heat equation, and a Kakutani fixed point argument. Keywords Time optimal control • Bang-bang property • Semilinear heat equation • Observability estimate from measurable sets • Switching control Mathematics Subject Classifications 35K58 • 49J20 • 49J30 • 93B07 Communicated by Viorel Barbu.

Relaxation schemes for mathematical programmes with switching constraints

“…The last q constraints in (MPSC) force G l (x) or H l (x) to be zero for all l ∈ Q, which gives rise to the terminology "switching constraints". Switching structures appear frequently in the context of optimal control, see Clason et al [2017], Gugat [2008], Hante and Sager [2013], Liberzon [2003], Seidman [2013], Wang and Yan [2015], Zuazua [2011], and the references therein, or as a reformulation of so-called either-or constraints, see [Mehlitz, 2018, Section 7]. Naturally, (MPSC) is related to other problem classes from disjunctive programming such as mathematical programs with complementarity constraints, MPCCs for short, see Luo et al [1996], Outrata et al [1998], or mathematical programs with vanishing constraints, MPVCs for short, see Achtziger and Kanzow [2008], Hoheisel and Kanzow [2008].…”

Optimality Conditions and Duality for MultiobjectiveSemi-infinite Optimization Problems with SwitchingConstraints on Hadamard Manifolds

“…The nomenclature 'switching constraints' indicates the fact that if the product of the two functions present in the mentioned equality constraint is equal to zero, atleast one of the functions must be zero. (MPSC) őnds its application in numerous őelds of modern research, in particular, in the őeld of optimal control (see, for instance, [15,45,61] and the references cited therein), where one of the control functions is often required to vanish at any instance of the underlying domain in case there are multiple control functions present (see, for instance, [17,9,10] and the references cited therein).…”

Optimality Conditions, Qualifications and Approximation Method for a Class of Non-Lipschitz Mathematical Programs with Switching Constraints

“…"Switching" means that the product of two functions is equal to 0 if and only if at least one of two functions equals 0. The current applications for switching structures appear in optimal control frequently [1][2][3] and optimization problems with either-or constraints [4]. In real world applications, switching structures naturally appear via modelling the bacteria switch instantaneously between active and dormant states [5], and modelling the state at the two boundary points to control a finite string to the zero state in finite time [6].…”