Exact boundary controllability on \(L_ 2(\Omega)\times H^{-1}(\Omega)\) of the wave equation with Dirichlet boundary control acting on a portion of the boundary \(\partial \Omega\), and related problems. (English) Zbl 0656.93011

Triggiani, R.

Appl. Math. Optimization 18, No. 3, 241-277 (1988).

This paper presents improvements to results on the exact controllability problem for the wave equation obtained by I. Lasiecka and the author [J. Differ. Equations 66, 340-390 (1987; Zbl 0629.93047)] via uniform stabilization and by J.-L. Lions [Contrôlabilité exacte, perturbations et stabilisation des systèmes distributés. Tome l (1988; Zbl 0653.93002)] via the HUM method. The problem consists in finding a Dirichlet boundary condition control on a part \(\Gamma_ 1\) of the boundary belonging to \(L^ 2(]0,T[\times \Gamma_ 1)\) that steers an arbitrary initial condition in \(L^ 2(\Omega)\times H^{-1}(\Omega)\) to zero. The key technical issue is a lower bound on the \(L^ 2(\Gamma_ 1\times]0,T[)\) norm of the normal derivative of the solution to the corresponding homogeneous problem.
This is done by the a priori estimate technique using a vector field which is more general than the radial vector field from an external point \(x_ 0\) which was utilized before. Conditions obtained on \(\Gamma_ 1\) are less restrictive and examples are given. The variable coefficient case is also studied.

Reviewer: J.Henry

MSC:

93B05	Controllability
35L05	Wave equation
93C20	Control/observation systems governed by partial differential equations
35B37	PDE in connection with control problems (MSC2000)
93B03	Attainable sets, reachability
93C05	Linear systems in control theory

Keywords:

exact controllability; wave equation; Dirichlet boundary condition control

Citations:

Zbl 0629.93047; Zbl 0653.93002

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