We consider the problem of placing a Dirichlet region made by n small balls of given radius in a given domain subject to a force f in order to minimize the compliance of the configuration. Then we let n tend to infinity and look for the Γ-limit of suitably scaled functionals, in order to get informations on the asymptotical distribution of the centres of the balls. This problem is both linked to optimal location and shape optimization problems.

In this paper we deal with the local exact controllability of theNavier-Stokes system with nonlinear Navier-slip boundaryconditions and distributed controls supported in small sets. In afirst step, we prove a Carleman inequality for the linearizedNavier-Stokes system, which leads to null controllability of thissystem at any time T>0. Then, fixed point arguments lead to thededuction of a local result concerning the exact controllabilityto the trajectories of the Navier-Stokes system.

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.

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∞ .

We study the Hamilton-Jacobi equation of the minimal time function in a domain which contains the target set. We generalize the results of Clarke and Nour [J. Convex Anal., 2004], where the target set is taken to be a single point. As an application, we give necessary and sufficient conditions for the existence of solutions to eikonal equations.

We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power $(-A)^\beta$ – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already been studied in the literature. But, due to the nonlinear term in the state equation, the same fractional power $(-A)^\beta$ appears in another nonlinear term whose behavior is different from the one of the Hamiltonian functional. We also consider cost functionals which are not bounded in bounded subsets in X, but only in bounded subsets in a space $Y\hookrightarrow X$ . To treat these new difficulties, we show that the value function of control problems we consider is equal in bounded sets in Y to the unique viscosity solution of some Hamilton-Jacobi-Bellman equation. We look for viscosity solutions in classes of functions which are Hölder continuous with respect to the time variable.

We consider a controllability problem for a beam, clamped at one boundary andfree at the other boundary, with an attached piezoelectric actuator. ByHilbert Uniqueness Method (HUM)and new results on diophantine approximations, weprove that the space of exactly initial controllable data depends on thelocation of the actuator. We also illustrate these results with numericalsimulations.

In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight $(1-\mu)$ is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop systemgenerates a C 0 group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors andgeneralized eigenvectors that forms a Riesz basis for the state Hilbert space.Furthermore, we show that when the weight $\mu>\frac{1}{2}$ , for any time delay,we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When $\mu=\frac{1}{2}$ , we show that the system is at most asymptotically stable. When $\mu<\frac{1}{2}$ , the system is always unstable.

We apply the well-known homotopy continuation method to address themotion planning problem (MPP) for smooth driftless control-affinesystems. The homotopy continuation method is a Newton-type procedureto effectively determine functions only defined implicitly. Thatapproach requires first to characterize the singularities of asurjective map and next to prove global existence for the solution ofan ordinary differential equation, the Wazewski equation. In thecontext of the MPP, the aforementioned singularities are the abnormalextremals associated to the dynamics of the control system and theWazewski equation is an o.d.e. on the control space called the PathLifting Equation (PLE). We first show elementary factsrelative to the maximal solution of the PLE such as local existence anduniqueness. Then we prove two general results, a finite-dimensionalreduction for the PLE on compact time intervals and aregularity preserving theorem. In a second part, if the Strong BracketGenerating Condition holds, we show, forseveral control spaces, the global existence of the solution of the PLE,extending a previous result of H.J. Sussmann.

An elliptic PDE is studied which is a perturbation of an autonomousequation. The existence of a nontrivial solution is proven viavariational methods. The domain of the equation is unbounded, whichimposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. Inan earlier paper with this assumption, a solution was obtained using asimple application of topological (Brouwer) degree. Here, a more subtledegree theory argument must be used.

This paper addresses the Cauchy problem for thegradient flow equation in a Hilbert space  $\mathcal{H}$ \[\begin{cases}u'(t)+ \partial_{\ell}\phi(u(t))\ni f(t)&\text{{\it a.e.}\ in }(0,T),u(0)=u_0, \end{cases}\] where $\phi: \mathcal{H} \to (-\infty,+\infty]$ is a proper,lower semicontinuous functional which is not supposed to be a(smooth perturbation of a) convex functional and $\partial_{\ell}\phi$ is(a suitable limiting version of) its subdifferential.We will present some new existence results for the solutions of theequation by exploiting a variational approximationtechnique, featuring some ideas from the theory of Minimizing Movementsand of Young measures.
Our analysisis also motivated by some models describing phase transitionsphenomena, leading tosystems of evolutionary PDEs which have a common underlying gradient flow structure:in particular, we will focus onquasistationary models, which exhibithighly non convex Lyapunov functionals.

This paper studies the attainable set at time T>0 for the control system $$\dot y(t)=f(y(t),u(t))\,\qquad u(t)\in U$$ showing that, under suitable assumptions on f, such a set satisfies a uniform interior sphere condition. The interior sphere property is then applied to recover a semiconcavity result for the valuefunction of time optimal control problems with a general target, and todeduce C1,1-regularity for boundaries of attainable sets.

Let $\Lambda $ be a Lagrangian submanifold of $T^{*}X$ for some closedmanifold X. Let $S(x,\xi )$ be a generating function for $\Lambda $ whichis quadratic at infinity, and let W(x) be the corresponding graph selectorfor $\Lambda ,$ in the sense of Chaperon-Sikorav-Viterbo, so that thereexists a subset $X_{0}\subset X$ of measure zero such that W is Lipschitzcontinuous on X, smooth on $X\backslash X_{0}$ and $(x,\partial W/\partialx(x))\in \Lambda $ for $X\backslash X_{0}.$ Let H(x,p)=0 for $(x,p)\in\Lambda$ . Then W is a classical solution to $H(x,\partial W/\partialx(x))=0$ on $X\backslash X_{0}$ and extends to a Lipschitz function on thewhole of X. Viterbo refers to W as a variational solution. We prove that W is also a viscosity solution under some simple and natural conditions.We also prove that these conditions are satisfied in many cases, includingcertain commonly occuring cases where H(x,p) is not convex in p.

A Lur'e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur'e system of equations. A sufficient strict circle criterion of solvability of the latter is found,which is based on results by Oostveen and Curtain [Automatica34 (1998) 953–967]. All theresults are illustrated in detail by an electrical transmission line example of thedistortionless loaded $\mathfrak{RLCG}$ -type. The paper uses extensively thephilosophy of reciprocal systems with bounded generating operators as recentlystudied and used by Curtain in (2003) [Syst. Control Lett.49 (2003) 81–89; SIAM J. Control Optim.42 (2003) 1671–1702].