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)).

We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spacesZ=L2(Ω)given byz′=−Az+1ωu(t),t∈[0,τ], whereΩis a domain inℝn,ωis an open nonempty subset ofΩ,1ωdenotes the characteristic function of the setω, the distributed controlu∈L2(0,t1;L2(Ω))andA:D(A)⊂Z→Zis an unbounded linear operator with the following spectral decomposition:Az=∑j=1∞λj∑k=1γj〈z,ϕj,k〉ϕj,k. The eigenvalues0<λ1<λ2<⋯<⋯λn→∞ofAhave finite multiplicityγjequal to the dimension of the corresponding eigenspace, and{ϕj,k}is a complete orthonormal set of eigenvectors ofA. The operator−Agenerates a strongly continuous semigroup{T(t)}given byT(t)z=∑j=1∞e−λjt∑k=1γj〈z,ϕj,k〉ϕj,k. Our result can be applied to thenD heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.

In this paper we give necessary and sufficient conditions for the approximate controllability of the following system of parabolic equations with delay:where Ω is a bounded domain in R N , D is an n × n nondiagonal matrix whose eigenvalues are semi-simple with nonnegative real part, the controlHere τ 0 is the maximum delay, which is supposed to be finite. We assume that the operator L : L 2 ([−τ , 0]; Z ) → Z is linear and bounded, and φ 0 ∈ Z , φ ∈ L 2 ([−τ , 0]; Z ). To this end: First, we reformulate this system into a standard first-order delay equation. Secondly, the semigroup associated with the first-order delay equation on an appropriate product space is expressed as a series of strongly continuous semigroups and orthogonal projections related with the eigenvalues of the Laplacian operator ( A = − ∂ ∂ 2 ); this representation allows us to reduce the controllability of this partial differential equation with delay to a family of ordinary delay equations. Finally, we use the well-known result on the rank condition for the approximate controllability of delay system to derive our main result.