Exact controllability of the suspension bridge model proposed by Lazer and McKenna

Abstract

In this paper we give a sufficient condition for the exact controllability of the following model of the suspension bridge equation proposed by Lazer and McKenna in [A.C. Lazer, P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev. 32 (1990) 537-578]:where t 0, d > 0, c > 0, k > 0, the distributed control u ∈ L 2 (0, t 1 ; L 2 (0, 1)), p : R × [0, 1] → R is continuous and bounded, and the non-linear term f : [0, t 1 ] × R × R → R is a continuous function on t and globally Lipschitz in the other variables, i.e., there exists a constant l > 0 such that for allTo this end, we prove that the linear part of the system is exactly controllable on [0, t 1 ]. Then, we prove that the non-linear system is exactly controllable on [0, t 1 ] for t 1 small enough. That is to say, the controllability of the linear system is preserved under the non-linear perturbation −kw + + p(t, x) + f (t, w, u(t, x)).

A Characterization of Semilinear Surjective Operators and Applications to Control Problems

“…Second, the results are so general that can be apply to those control systems governed by evolutions equations like the one studied in [1][2][3] and [4]. Third, we find a formula for a control steering the system from the initial state 0 z to a final state 1 z on time 0   , for both the linear and the nonlinear systems, which is very important from engineering point of view.…”

“…where ( ) z   is the solution of (4.14) corresponding to the control u define by: z is given by: 1 z is given by:…”

“…A physical interpretation of the nonlinear term ( ) t f t u w w    could be as an eternal force like in the suspension bridge equation proposed by Lazer and McKenna (see [1]). …”

Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space

“…has an induced inverse operator W −1 which takes values in L F 2 (J, U ) \ Ker W (see [8,21] for an example of e −At a C 0 semigroup satisfying the condition) and there exist positive constants M 3 , M 4 such that H )) is an L 2 -Carathéodory function; (H5) there exists a continuous nondecreasing function ψ : R + → (0, ∞), P ∈ L 1 (J, R + ) such that…”

Controllability of Semilinear Systems of Parabolic Equations with Delay on the State

“…"> H2. f is a smooth enough function. Remark . The function f is smooth enough if: The Solution z ( u ) = z u of the problem is unique. The solution z ( u ) = z u of the problem depends continuously on u . …”