Optimal control of an energy-critical semilinear wave equation in 3D with spatially integrated control constraints

Abstract

This paper is concerned with an optimal control problem subject to the H 1 -critical defocusing semilinear wave equation on a smooth and bounded domain in three spatial dimensions. Due to the criticality of the nonlinearity in the wave equation, unique solutions to the PDE obeying energy bounds are only obtained in special function spaces related to Strichartz estimates and the nonlinearity. The optimal control problem is complemented by pointwise-in-time constraints of Trust-Region type u(t) L 2 (Ω) ≤ ω(t). We prove existence of globally optimal solutions to the optimal control problem and give optimality conditions of both first-and second order necessary as well as second order sufficient type. A nonsmooth regularization term for the natural control space L 1 (0, T; L 2 (Ω)), which also promotes sparsity in time of an optimal control, is used in the objective functional.

Turnpike in Lipschitz—nonlinear optimal control

“…the Burgers equation); along these lines we refer to [49] for a local turnpike result for the 2D Navier-Stokes system. Similar questions can be asked for the semilinear wave equation, where the nonlinearity is sometimes only assumed to be superlinear (see [27] for a subcritical optimal control study)-our methodology a priori applies if the nonlinearity is either truncated by some cut-off, or if one manages to prove uniform estimates of y T L ∞ ((0,T)×Ω) with respect to T. Further nonlinear problems which could be investigated include hyperbolic systems (see [22] for a related study) or free boundary problems (see [17] for a control perspective).…”

A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation

“…Note that the space V appears commonly in optimal control problems with wave equations; cf. e. g. [69] and the references therein.…”

“…Optimal control of semilinear wave equations and more general hyperbolic equations is in itself already very challenging, and such problems have been in the focus of research interest for a long time. A selection of notable publications include articles from Ismayilova [63], Kunisch and Meinlschmidt [69], Kunisch, Trautmann, and Vexler [70], Pfaff and Ulbrich [98], Schmitt and Ulbrich [106], and Zuazua [130]. Related research in the narrower context of hyperbolic equations describing the gas flow through pipes, in particular Euler equations, is due to Gugat, Dick, and Leugering [52] and Gugat and Ulbrich [53,54]; see also [56,57].…”