On the uniform controllability for a family of non-viscous and viscous Burgers-α systems

In this paper we study the global controllability of families of the so called non-viscous and viscous Burgers-α systems by using boundary and space independent distributed controls. In these equations, the usual convective velocity of the Burgers equation is replaced by a regularized velocity, induced by a Helmholtz filtered of characteristic wave-length α. First, we prove a global exact controllability result (uniform with respect to α) for the non-viscous Burgers-α system, using the return method and a fixed-point argument. Then, the global uniform exact controllability to constant states is deduced for the viscous equations. To this purpose, we first prove a local exact controllability property and, then, we establish a global approximate controllability result for smooth initial and target states.

Exact null-controllability of interconnected abstract evolution equations with unbounded input operators

<p style='text-indent:20px;'>The exact null-controllability problem in the class of smooth controls with applications to interconnected systems was considered in [<xref ref-type="bibr" rid="b23">23</xref>] for the case of bounded input operators appearing in systems with distributed controls. The current paper constitutes an extension of the [<xref ref-type="bibr" rid="b23">23</xref>] for the case of unbounded input operators (with more emphasis on the controllability of interconnected systems). The proofs of the results of [<xref ref-type="bibr" rid="b23">23</xref>] for the case of bounded input operators are adopted for the case of unbounded input operators.</p>

Optimal distributed and tangential boundary control for the unsteady stochastic Stokes equations

We consider a control problem constrained by the unsteady stochastic Stokes equations with nonhomogeneous boundary conditions in connected and bounded domains. In this paper, controls are defined inside the domain as well as on the boundary. Using a stochastic maximum principle, we derive necessary and sufficient optimality conditions such that explicit formulas for the optimal controls are derived. As a consequence, we are able to control the stochastic Stokes equations using distributed controls as well as boundary controls in a desired way.

Internal null controllability of the generalized Hirota-Satsuma system

The generalized Hirota-Satsuma system consists of three coupled nonlinear Korteweg-de Vries (KdV) equations. By using two distributed controls it is proven in this paper that the local null controllability property holds when the system is posed on a bounded interval. First, the system is linearized around the origin obtaining two decoupled subsystems of third order dispersive equations. This linear system is controlled with two inputs, which is optimal. This is done with a duality approach and some appropriate Carleman estimates. Then, by means of an inverse function theorem, the local null controllability of the nonlinear system is proven.

Non null controllability of Stokes equations with memory

In this paper, we consider the null controllability problem for the Stokes equations with a memory term. For any positive final time T > 0, we construct initial conditions such that the null controllability does not hold even if the controls act on the whole boundary. We also prove that this negative result holds for distributed controls.