Bilinear optimal control of the velocity term in a Kirchhoff plate equation. (English) Zbl 0936.49003

Bradley, Mary Elizabeth; Lenhart, Suzanne; Yong, Jiongmin

J. Math. Anal. Appl. 238, No. 2, 451-467 (1999).

The authors consider the bilinear control of a vibrating Kirchhoff plate. The \(L^\infty\) control is the dual of velocity, implying that its physical dimension is that of momentum. The state equation is: \(w_{tt}+ \Delta^2w= h(t)w_t\) on \(Q= \Omega\times [0,T]\), with prescribed initial velocity and displacement. Both displacement and normal component of velocity are set to zero on the boundary segment \(\Gamma_0\) of the region \(\Omega\) occupied by the plate. The plate is clamped on the part of boundary denoted by \(\Gamma_0\), implying \(w=\partial w/\partial\nu=0\), while it is free on the remainder of the boundary denoted by \(\Gamma_1\). Let \(\nu= \{\nu_1,\nu_2\}\) be the outward pointing normal vector on the boundary, while \(\tau\) is the tangential unit vector. Then state equations on \(\Gamma_1\) are: \(\Delta w+(1- \nu)B_1w= 0\), and \(\partial\Delta w/\partial\nu+ (1-\mu)(1-\nu)B_2w= 0\), where \(B_1w= 2n_1n_2 w_{xy}- n^2_1 w_{yy}- n^2_2 w_{xx}\) and \(B_2w= \partial/\partial\tau[(n^2_1- n^2_2) w_{xy}+ n_1n_2(w_{yy}- w_{xx})]\). \(\mu\) is the Poisson ratio. The cost functional is \(J= \left\{\int_Q(w- z)^2dQ+ \beta_0\int^T h^2(t) dt\right\}\), where \(z\) is the desired evolution of the vibrating plate, while the second integral is the cost of implementing the control. The optimal control \(h^*\) minimizes \(J\). The standard bilinear form \(a(u,v)\) generates the norm equivalent to the usual \(H^2\) norm. The weak solution is located in an appropriate Sobolev space, and the state equation is rewritten in a semigroup form: \(Aw= \Delta^2w\), where \(D(A)\) is carefully defined. Then \(\widehat A= \left[\begin{smallmatrix} 0 & I\\ A & 0\end{smallmatrix}\right]\) and the state equation becomes: \(d/dt(\widehat w(t))= \widehat A\widehat w+ B\widehat w\), with \(\widehat w(0)= \widehat w_0= [w_0, w_1]^T\). Operator \(\widehat A\) generates a strongly continuous semigroup on \(H\). A fixed point argument is used to prove the existence of a weak solution. Existence of smooth approximations follow from these arguments and from Gronwall’s inequality. A fairly easy proof of existence of optimal control uses well-known Egorov’s theorem asserting uniform convergence (except on a set of arbitrary small measure) which implies convergence of a sequence of cost functionals for approximate controls to the minimizing functional corresponding to the optimal control. Several necessary conditions for the existence of an optimal admissible control are proved. Uniqueness of solutions is proved following the assumption that the time interval is sufficiently small. This assumption of short time interval turns up in many problems of hyperbolic equations and in mixed-type problems of vibrating plates and shells. It probably cannot be weakened. The article is well-written, and well-organized. Detailed proofs are given of all assertions.

Reviewer: Vadim Komkov (Florida)

MSC:

49J20	Existence theories for optimal control problems involving partial differential equations
74M05	Control, switches and devices (“smart materials”) in solid mechanics
35L20	Initial-boundary value problems for second-order hyperbolic equations
35Q72	Other PDE from mechanics (MSC2000)

Keywords:

existence; uniqueness; vibrating Kirchhoff plate; optimal control

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