In this paper, we prove the genericity of the observability for discrete-time systems with more outputs than inputs.

A model representing the vibrations of a fluid-solid coupled structure is considered. Following Hilbert Uniqueness Method (HUM) introduced by Lions, we establish exact controllability results for this model with an internal controlin the fluid part and there is no control in the solid part. Novel features which arise because of the coupling are pointed out. It is a source of difficulty in the proof of observability inequalities, definition of weak solutions and the proof of controllability results.

In this work we are interested in the study of controllability andstabilization of the linearized Benjamin-Ono equation withperiodic boundary conditions, which is a generic model forthestudy of weakly nonlinear waves with nonlocal dispersion. It iswell known that theBenjamin-Ono equation has infinite number ofconserved quantities, thus we consider only controls acting in theequation such that the volume of the solution is conserved. Westudy also the stabilization with a feedback law which gives us anexponential decay of the solutions.

Sliding mode control of ordinary differential equations isconsidered. A key robustness property, called approximability, is studied froman optimization point of view. It is proved that Tikhonov well-posedness of asuitably defined optimization problem is intimately related to approximability.Making use of this link, new approximability criteria are obtained for nonlinear sliding modecontrol systems.

We consider an Hamilton-Jacobi equation of the form

 $$ H(x,Du)=0\quad x\in\Omega\subset\mathbb R^N,\qquad\qquad (1) $$ 
where H(x,p) is assumed Borel measurable and quasi-convex inp. The notion of Monge solution, introduced by Newcomb and Su,is adapted to this setting making use of suitable metric devices.We establish the comparison principle for Monge sub andsupersolution, existence and uniqueness for equation ([see full text])coupled with Dirichlet boundary conditions, and a stability result. Therelation among Monge and Lipschitz subsolutions is also discussed.

In this paper, we consider a class of stochasticmathematical programs with equilibrium constraints (SMPECs) thathas been discussed by Lin and Fukushima (2003). Based on areformulation given therein, we propose a regularization methodfor solving the problems. We show that, under a weak condition, anaccumulation point of the generated sequence is a feasible pointof the original problem. We also show that such an accumulationpoint is S-stationary to the problem under additional assumptions.

We study the homogenization of parabolic or hyperbolic equations like \[\rho_\varepsilon{\partial^n u_\varepsilon\over \partial t^n}- {\rm div}(a_\varepsilon\nabla u_\varepsilon) =f \ \ \hbox{ in } \quad {\O\times(0,T)}+\ \ \hbox{\rm boundary conditions}, \quad n \in \{1,2\},\] when the coefficients $\rho_\varepsilon$ , $a_\varepsilon$ (defined in Ω) take possibly high values on a ε-periodic set of grain-like inclusions of vanishing measure.Memory effects arise in the limit problem.

This paper concerns constrained dynamic optimization problemsgoverned by delay control systems whose dynamic constraints are described by bothdelay-differential inclusions and linear algebraic equations. This is a new class ofoptimal control systems that, on one hand, may be treated as a specific type ofvariational problems for neutral functional-differential inclusions while, on the otherhand, is related to a special class of differential-algebraic systems with a generaldelay-differential inclusion and a linear constraint link between “slow” and “fast”variables. We pursue a twofold goal: to study variational stability for this class ofcontrol systems with respect to discrete approximations and to derive necessaryoptimality conditions for both delayed differential-algebraic systems under considerationand their finite-difference counterparts using modern tools of variational analysis andgeneralized differentiation. The authors are not familiar with any results in thesedirections for such systems even in the delay-free case. In the first part of the paperwe establish the value convergence of discrete approximations as well as the strongconvergence of optimal arcs in the classical Sobolev space W -1,2. Then using discreteapproximations as a vehicle, we derive necessary optimality conditions for the initialcontinuous-time systems in both Euler-Lagrange and Hamiltonian forms via basicgeneralized differential constructions of variational analysis.