We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to the control which consists of horizontal moves. We assume that the motion of the fluid is well-described by the Saint–Venant equations (also called the shallow water equations). We prove the local controllability of this nonlinear control system around any steady state. As a corollary we get that one can move from any steady state to any other steady state.

One proves that the steady-state solutions to Navier–Stokes equations with internal controllers are locally exponentially stabilizable by linear feedback controllers provided by a LQ control problem associated with the linearized equation.

This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently formally passing to the limit of continuous time. The resulting functionals may be regarded as aweighted dissipation-energy functional with a weight decaying with a rate $1/\epsilon$ . The corresponding Euler-Lagrange equation leads to anelliptic regularization of the original evolutionary problem. The Γ-limit of these functionals for $\epsilon\to 0$ is highlydegenerate and provides limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. Under general assumptions on the functionals, we show that all such limits satisfy the energetic formulation (S) & (E) for rate-independent systems. Moreover, we show that the accumulation points of the regularized solutions solve the associated limiting energetic formulation.

The numerical resolution of the low thrust orbital transfer problem around the Earth with the maximization of the final mass or minimization of the consumption is investigated. This problem is difficult to solve by shooting method because the optimal control is discontinuous and a homotopic method is proposed to deal with these difficulties for which convergence properties are established. For a thrust of 0.1 Newton and a final time 50% greater than the minimum one, we obtain 1786 switching times.

The receding horizon control strategy fordynamical systems posed in infinite dimensional spaces is analysed. Itsstabilising property is verified provided controlLyapunov functionals are used as terminal penalty functions.For closed loop dissipative systems the terminal penalty canbe chosen as quadratic functional. Applications to the Navier–Stokesequations, semilinear wave equations and reaction diffusion systems are given.

Several recent results in the area of robust asymptotic stability of hybrid systems show that the concept of a generalized solution to a hybrid system is suitable for the analysis and design of hybrid control systems. In this paper, we show that such generalized solutions are exactly the solutions that arise when measurement noise in the system is taken into account.

We define and characterize weak and strong two-scale convergence in Lp ,C 0 and other spaces via a transformation of variable, extending Nguetseng's definition. We derive several properties, including weak and strong two-scale compactness;in particular we prove two-scale versions of theorems of Ascoli-Arzelà, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions.We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey.

In this paper we introduce a numerical approach adapted to the minimizationof the eigenmodes of a membrane with respect to the domain. This method isbased on the combination of the Level Set method of S. Osher and J.A.Sethian with the relaxed approach. This algorithm enables both changing thetopology and working on a fixed regular grid.

We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is well-known that this problem is highly illposed. In this work, we propose the use of the Mumford–Shah functional, developed for segmentation and denoising of images, as a regularization. After establishing existence properties of the resulting variational problem, we proceed by demonstrating the approach in several numerical examples. Our results indicate that this is an effective approach for overcoming the illposedness. Moreover, it has the capability of enhancing the reconstruction while at the same time segmenting the conductivity image.

The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional and a numerical application is presented.

The goal of this paper is to derive some error estimates for thenumerical discretization of some optimal control problems governedby semilinear elliptic equations with bound constraints on thecontrol and a finitely number of equality and inequality stateconstraints. We prove some error estimates for the optimalcontrols in the L ∞ norm and we also obtain error estimatesfor the Lagrange multipliers associated to the state constraintsas well as for the optimal states and optimal adjoint states.

This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of systems (among them converse Lyapunov results). The proposed framework can allow the study of discontinuous solutions for nonlinear systems described by a single first-order hyperbolic partial differential equation under the effect of measurable inputs acting on the boundary and/or on the differential equation. Illustrative examples show that the conversion of a system described by a single first-order hyperbolic partial differential equation to an integral delay system can simplify considerably the stability analysis and the solution of robust feedback stabilization problems.

We describe an algorithm for computing the value function for “all source, single destination” discrete-time nonlinear optimal control problems together with approximations of associated globally optimal control strategies. The method is based on a set oriented approach for the discretization of the problem in combination with graph-theoretic techniques. The central idea is that a discretization of phase space of the given problem leads to an (all source, single destination) shortest path problem on a finite graph. The method is illustrated by two numerical examples, namely a single pendulum on a cart and a parametrically driven inverted double pendulum.

In this paper sufficient optimality conditions are established for optimal control ofboth steady-state and instationary Navier-Stokes equations. The second-order condition requirescoercivity of the Lagrange function on a suitable subspace together with first-order necessaryconditions. It ensures local optimality of a reference function in a Ls -neighborhood, whereby the underlying analysis allows to use weaker norms than L∞ .

In this paper we study a controllability problem for a simplified one dimensional model for the motion of a rigid body in a viscous fluid. The control variable is the velocity of the fluid at one end. One of the novelties brought in with respect to the existing literature consists in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem. This methodology is based on an abstract argument for the null controllability of parabolic equations in the presence of source terms and it avoids tackling linearized problems with time dependent coefficients.

We study a class of hyperbolic partial differential equations on a one dimensional spatial domain with control and observation at the boundary. Using the idea of feedback we show these systems are well-posed in the sense of Weiss and Salamon if and only if the state operator generates a C 0-semigroup. Furthermore, we show that the corresponding transfer function is regular, i.e., has a limit for s going to infinity.

In this paper, we establish Carleman estimates for the twodimensional isotropic non-stationary Lamé system with the zero Dirichlet boundaryconditions. Using this estimate, we prove the uniqueness and thestability in determining spatially varying density and two Lamécoefficients by a single measurement of solution over (0,T) x ω, where T > 0 is a sufficiently large time interval and a subdomainω satisfies a non-trapping condition.

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized.Lower and upper bounds on the first conjugate time are proved.The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.

The concepts of stability, attractivity and asymptotic stability for systems subject to restrictions of the input values are introduced and analyzed in terms of Lyapunov functions. A comparisonwith the well known input-to-state stability property introduced by Sontag is provided. We use these concepts in order to derive sufficient conditions for global stabilization for triangular andfeedforward systems by means of saturated bounded feedback controllers and also recover some recent results due to Teel.

In this paper we are concerned with questions of multiplicity and concentration behaviorof positive solutions of the elliptic problem

$$ (P_{\var})\hspace*{4cm} \left\{ \begin{array}{rcl} \mathcal{L}_{\var}u=f(u) \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u>0 \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u \in H^{1}(\R^3), \end{array} \right. $$$\mathrm{\left(}{\mathit{P}}_{\mathit{\epsilon }}\mathrm{\right)}\left\{\begin{array}{c}\\ {\mathrm{ℒ}}_{\mathit{\epsilon }}\mathit{u}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathit{>}\mathrm{0}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathrm{\in }{\mathit{H}}^{\mathrm{1}}\mathrm{\left(}{\mathrm{IR}}^{\mathrm{3}}\mathrm{\right)}\mathit{,}\end{array}\right.$

where ε is a small positiveparameter, f : ℝ → ℝ is a continuous function, $$ \mathcal{L}_{\var} $$${\mathrm{ℒ}}_{\mathit{\epsilon }}$ is a nonlocaloperator defined by

$$ \mathcal{L}_{\var}u=M\left(\dis\frac{1}{\var}\int_{\R^{3}}|\nabla u|^{2}+\frac{1}{\var^{3}}\dis\int_{\R^{3}}V(x)u^{2}\right)\left[-\var^{2}\Delta u + V(x)u \right], $$${\mathrm{ℒ}}_{\mathit{\epsilon }}\mathit{u}\mathrm{=}\mathit{M}\left(\frac{\mathrm{1}}{\mathit{\epsilon }}{\mathrm{\int }}_{{\mathrm{IR}}^{\mathrm{3}}}\mathrm{|}\mathrm{\nabla }\mathit{u}{\mathrm{|}}^{\mathrm{2}}\mathrm{+}\frac{\mathrm{1}}{{\mathit{\epsilon }}^{\mathrm{3}}}{\mathrm{\int }}_{{\mathrm{IR}}^{\mathrm{3}}}\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathrm{2}}\right)\mathrm{[}\mathrm{-}{\mathit{\epsilon }}^{\mathrm{2}}\mathrm{\Delta }\mathit{u}\mathrm{+}\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathrm{]}}\mathit{,}$

M : IR+ → IR+ andV : IR3 → IR are continuous functions whichverify some hypotheses.