An approximation procedure for time optimal control problems for the linear wave equation is analyzed. Its asymptotic behavior is investigated and an optimality system including the maximum principle and the transversality conditions for the regularized and unregularized problems are derived.

The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.

In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin andGhidaglia for the Korteweg-de Vries equation posed on a bounded domain(0,L). We show that this class of Initial-Boundary Value Problems islocally well-posed in the classical Sobolev spaceHs(0,L) for s > -3/4, which provides a positive answer to one of the openquestions of Colin and Ghidaglia [Adv. Differ. Equ. 6 (2001)1463–1492].

The least Steklov eigenvalue d1 for the biharmonic operatorin bounded domains gives a bound for the positivity preserving property for the hingedplate problem, appears as a norm of a suitable trace operator, and gives the optimalconstant to estimate the L2-norm of harmonic functions. Theseapplications suggest to address the problem of minimizing d1in suitable classes of domains. We survey the existing results and conjectures about thistopic; in particular, the existence of a convex domain of fixed measure minimizingd1 is known, although the optimal shape is still unknown. Weperform several numerical experiments which strongly suggest that the optimal planar shapeis the regular pentagon. We prove the existence of a domain minimizingd1 also among convex domains having fixed perimeter andpresent some numerical results supporting the conjecture that, among planar domains, thedisk is the minimizer.

A two-person zero-sum differential game with unbounded controls is considered. Underproper coercivity conditions, the upper and lower value functions are characterized as theunique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacsequations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upperand lower value functions coincide, leading to the existence of the value function of thedifferential game. Due to the unboundedness of the controls, the corresponding upper andlower Hamiltonians grow super linearly in the gradient of the upper and lower valuefunctions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobiequations involving such kind of Hamiltonian is proved, without relying on theconvexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivityconditions guaranteeing the finiteness of the upper and lower value functions are sharp insome sense.

We consider the problem of minimising the nth-eigenvalue of the RobinLaplacian in RN. Although for n = 1,2 and apositive boundary parameter α it is known that the minimisers do notdepend on α, we demonstrate numerically that this will not always be thecase and illustrate how the optimiser will depend on α. We derive aWolf–Keller type result for this problem and show that optimal eigenvalues grow at mostwith n1/N, which is in sharp contrast withthe Weyl asymptotics for a fixed domain. We further show that the gap between consecutiveeigenvalues does go to zero as n goes to infinity. Numerical results thensupport the conjecture that for each n there exists a positive value ofαn such that the ntheigenvalue is minimised by n disks for all0 < α < αnand, combined with analytic estimates, that this value is expected to grow withn1/N.

Time optimal control problems for an internally controlled heat equation with pointwisecontrol constraints are studied. By Pontryagin’s maximum principle and properties ofnontrivial solutions of the heat equation, we derive a bang-bang property for time optimalcontrol. Using the bang-bang property and establishing certain connections between timeand norm optimal control problems for the heat equation, necessary and sufficientconditions for the optimal time and the optimal control are obtained.

We compute the Γ-limit of a sequence of non-local integral functionalsdepending on a regularization of the gradient term by means of a convolution kernel. Inparticular, as Γ-limit, we obtain free discontinuity functionals withlinear growth and with anisotropic surface energy density.

We apply four different methods to study an intrinsically bang-bang optimal control problem. We study first a relaxed problem that we solve with a naive nonlinear programming approach. Since these preliminary results reveal singular arcs, we then use Pontryagin’s Minimum Principle and apply multiple indirect shooting methods combined with homotopy approach to obtain an accurate solution of the relaxed problem. Finally, in order to recover a purely bang-bang solution for the original problem, we use once again a nonlinear programming approach.

In a recent article [B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 1081–1098], we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is g = dϕ2 + m(ϕ)dθ2 to the period mapping of the ϕ-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics. These properties have applications in optimal control of space and quantum mechanics, and in optimal transport.

A lower semicontinuity result in BV is obtained for quasiconvexintegrals with subquadratic growth. The key steps in this proof involve obtainingboundedness properties for an extension operator, and a precise blow-up technique thatuses fine properties of Sobolev maps. A similar result is obtained by Kristensen in[Calc. Var. Partial Differ. Equ. 7 (1998) 249–261], wherethere are weaker asssumptions on convergence but the integral needs to satisfy a strongergrowth condition.

This paper deals with the existence of solutions to the following system:

$$\left\{\begin{array}{l} -\Deltau+u=\frac{\alpha}{\alpha+\beta}a(x)|v|^{\beta} |u|^{\alpha-2}u\quad\mbox{ in}\mathbb{R}^N\\ [0.2cm] -\Delta v+v=\frac{\beta}{\alpha+\beta}a(x)|u|^{\alpha}|v|^{\beta-2}v\quad\mbox{ in }\mathbb{R}^N. \end{array}\right.$$$\left\{\begin{array}{c}\\ \mathrm{-}\mathit{\Delta u}\mathrm{+}\mathit{u}\mathrm{=}\frac{\mathit{\alpha }}{\mathit{\alpha }\mathrm{+}\mathit{\beta }}\mathit{a}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\left|}\mathit{v}{\mathrm{|}}^{\mathit{\beta }}\mathrm{\right|}\mathit{u}{\mathrm{|}}^{\mathit{\alpha }\mathrm{-}\mathrm{2}}\mathit{u}\mathrm{in}{R}^{\mathit{N}}\\ \mathrm{-}\mathit{\Delta v}\mathrm{+}\mathit{v}\mathrm{=}\frac{\mathit{\beta }}{\mathit{\alpha }\mathrm{+}\mathit{\beta }}\mathit{a}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\left|}\mathit{u}{\mathrm{|}}^{\mathit{\alpha }}\mathrm{\right|}\mathit{v}{\mathrm{|}}^{\mathit{\beta }\mathrm{-}\mathrm{2}}\mathit{v}\mathrm{in}{R}^{\mathit{N}}\mathit{.}\end{array}\right.$

With the help of the Nehari manifold and the linking theorem, we prove the existence ofat least two nontrivial solutions. One of them is positive. Our main tools are theconcentration-compactness principle and the Ekeland’s variational principle.

We study an optimal boundary control problem for the two dimensional unsteady linearized compressible Navier–Stokes equations in a rectangle. The control acts through the Dirichlet boundary condition. We first establish the existence and uniqueness of the solution for the two-dimensional unsteady linearized compressible Navier–Stokes equations in a rectangle with inhomogeneous Dirichlet boundary data, not necessarily smooth. Then, we prove the existence and uniqueness of the optimal solution over the control set. Finally we derive an optimality system from which the optimal solution can be determined.

We prove pointwise gradient bounds for entire solutions of pde’s of the form

     ℒu(x) = ψ(x, u(x), ∇u(x)),

where ℒ is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.