This paper discusses the properties of reachability and observability for linear systems over the max-plus algebra. Working in the event-domain, the concept of asticity is used to develop conditions for weak reachability and weak observability. In the reachability problem, residuation is used to determine if a state is reachable and to generate the required control sequence to reach it. In the observability problem, residuation is used to estimate the state. Finally, as in the continuous-variable...

We show that the set of nonnegative equilibrium-like states, namely, like $(y_d,0)$ of the semilinear vibrating string that can be reached from any non-zero initial state $(y_0,y_1)\in H_0^1(0,1)\times L^2(0,1)$, by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace $L^2(0,1)\times \left\{0\right\}$ of $L^2(0,1)\times H^{-1}(0,1)$. Our main results deal with nonlinear terms which admit at most the linear growth at infinity in $y$ and satisfy certain restriction on their total impact on (0,∞) with respect to the time-variable.

This paper describes the controllability of nonlinear fractional delay integrodifferential systems with multiple delays in control. Necessary and sufficient conditions for the controllability criteria for linear fractional delay system are established. Further sufficient conditions for the controllability of nonlinear fractional delay integrodifferential system are obtained by using fixed point arguments. Examples are provided to illustrate the results.

We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain $\Omega$ with control distributed in a subdomain $\omega \subset \Omega \subset {ℝ}^n,n\in \{2,3\}$. The result that we obtained in this paper is as follows. Suppose that $\widehat{v}(t,x)$ is a given solution of the Navier-Stokes equations. Let $v_0\left(x\right)$ be a given initial condition and $\parallel \widehat{v}(0,·)-v_0\parallel \<\epsilon$ where $\epsilon$ is small enough. Then there exists a locally distributed control $u,\text{supp}u\subset (0,T)\times \omega$ such that the solution $v(t,x)$ of the Navier-Stokes equations:$${\partial }_{t}v-\Delta v+(v,\nabla )v=\nabla p+u+f,\text{div}v=0,v{{|}_{\partial \Omega }=0,v|}_{t=0}=v_0$$coincides with...

We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain Ω with control distributed in a subdomain $\omega \subset \Omega \subset {ℝ}^n,n\in \{2,3\}$. The result that we obtained in this paper is as follows. Suppose that $\widehat{v}(t,x)$ is a given solution of the Navier-Stokes equations. Let $v_0\left(x\right)$ be a given initial condition and $\parallel \widehat{v}(0,·)-v_0\parallel <\epsilon$ where ε is small enough. Then there exists a locally distributed control $u,\text{supp}u\subset (0,T)\times \omega$ such that the solution v(t,x) of the Navier-Stokes equations: $${\partial }_{t}v-\Delta v+(v,\nabla )v=\nabla p+u+f,\text{div}v=0,v{{|}_{\partial \Omega }=0,v|}_{t=0}=v_0$$ coincides...

In this paper we deal with the null controllability problem for the heat equation with a memory term by means of boundary controls. For each positive final time T and when the control is acting on the whole boundary, we prove that there exists a set of initial conditions such that the null controllability property fails.

We briefly present the difficulties arising when dealing with the controllability of the discrete wave equation, which are, roughly speaking, created by high-frequency spurious waves which do not travel. It is by now well-understood that such spurious waves can be dealt with by applying some convenient filtering technique. However, the scale of frequency in which we can guarantee that none of these non-traveling waves appears is still unknown in general. Though, using Hautus tests, which read the...

In this paper we study a control problem for elliptic nonlinear monotone problems with Dirichlet boundary conditions where the control variables are the coefficients of the equation and the open set where the partial differential problem is studied.