In this article we apply the optimal andthe robust control theory to the sine-Gordon equation. In our casethe control is given by the boundary conditions and we work in a finitetime horizon. We present at the beginning the optimal control problemand we derive a necessary condition of optimality and we continue byformulating a robust control problem for which existence and uniquenessof solutions are derived.

In this paper, we prove someregularity results for the boundary of an open subset of $\xR^d$ whichminimizes the Dirichlet's energy among all open subsets withprescribed volume. In particular we show that, whenthe volume constraint is “saturated”,the reduced boundary of the optimal shape (and even the wholeboundary in dimension 2)is regular if the state function is nonnegative.

We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*Mas a symplectic manifold.The definition of “minsol solution” is suited to proving regularity results about u; in particular, we provein the first part that the closure of the set where u is not regular may be covered by a countable number of $n-1$ dimensional manifolds, but for a ${{\mathcal H}}^{n-1}$ negligeable subset.These results can be applied to the cutlocus of a C 2 submanifold of a Finsler manifold.

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.

Turnpike theorems deal with the optimality of trajectories reaching asingular solution, in calculus of variations oroptimal control problems.For scalar calculus of variations problems in infinite horizon, linear withrespect to the derivative, we use the theory of viscosity solutions ofHamilton-Jacobi equations to obtain a unique characterization of the valuefunction.With this approach, we extend for the scalar case the classical result based onGreen theorem, when there is uniqueness of the singular solution.We provide a new necessary and sufficient condition for turnpikeoptimality, even in the presence of multiple singular solutions.

A distributed optimal control problem for evolutionary Stokes flows isstudied via a pseudocompressibility formulation. Several results concerning the analysis of the velocity tracking problem are presented. Semidiscrete finite element error estimates for the correspondingoptimality system are derived based on estimates for the penalizedStokes problem and the BRR (Brezzi-Rappaz-Raviart) theory. Finally, theconvergence of the solutions of the penalized optimality systems as ε → 0 is examined.

We study a finite horizon problem for a system whose evolution isgoverned by a controlled ordinary differential equation, which takesalso account of a hysteretic component: namely, the outputof a Preisach operator of hysteresis. We derive a discontinuousinfinitedimensional Hamilton–Jacobi equation and prove that, under fairlygeneral hypotheses, the value function is the unique bounded anduniformly continuous viscosity solution of the corresponding Cauchyproblem.

A homogenization problem related to the micromagnetic energy functional is studied. In particular, the existence of the integral representation for the homogenized limit of a family of energies $$ \mathcal{E}_{\varepsilon}(m)=\int_{\Omega} \phi\left(x,\frac{x}{\varepsilon},m(x)\right)\,{\rm d}x-\int_{\Omega}h_e(x)\cdot m(x)\,{\rm d}x+\frac{1}{2}\int_{\mathbb R^3}|\nabla u(x)|^2\,{\rm d}x$$ of a large ferromagnetic body is obtained.

We compute numerically the minimizers of the Dirichlet energy $$E(u)=\frac{1}{2}\int_{B^2}|\nabla u|^2 {\rm d}x$$ among maps $u:B^2\to S^2$ from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P 1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned projected gradient). In order to improve the convergence, we generalize to manifolds the classical Newton and conjugate gradient algorithms. We give a proof of the quadratic convergence of the Newton algorithm for manifolds in a general setting.

We consider an optimal control problem describing a laser-inducedpopulation transfer on a n-level quantum system. For a convex cost depending only on the moduliof controls (i.e. the lasers intensities),we prove that there always exists a minimizer inresonance. This permits to justifysome strategies used in experimental physics. It is also quite importantbecause it permits to reduce remarkablythe complexity of the problem (and extend some of our previous resultsfor n=2 and n=3): instead of looking for minimizers on thesphere $S^{2n-1}\subset\mathbb{C}^n$ one is reduced to look just forminimizers on the sphere $S^{n-1}\subset \mathbb{R}^n$ . Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.

A hybrid flexible beam equation with harmonicdisturbance at the end where a rigid tip body is attached isconsidered. A simple motor torque feedback control is designedfor which only the measured time-dependent angle of rotation andits velocity are utilized. It is shown that this control can impelthe amplitude of the attached rigid tip body tending to zero astime goes to infinity.

In the 1950's and 1960's surface physicists/metallurgists such as Herring and Mullins applied ingenious thermodynamic arguments to explain a number of experimentally observed surface phenomena in crystals. These insights permitted the successful engineering of a large number of alloys, where the major mathematical novelty was that the surface response to external stress was anisotropic. By examining step/terrace (vicinal) surface defects it was discovered through lengthy and tedious experiments that the stored energy density (surface tension) along a step edge was a smooth symmetric function β of the azimuthal angle θ to the step, and that the positive function β attains its minimum value at $\theta = \pi/2$ and its maximum value at $\theta = 0$ . The function β provided the crucial thermodynamic parameters needed for the engineering of these materials. Moreover the minimal energy configuration of the step is determined by the values of the stiffness function $\beta'' + \beta$ which ultimately leads to the magnitude and direction of surface mass flow for these materials. In the 1990's there was a dramatic improvement in electron microscopy which permitted real time observation of the meanderings of a step edge under Brownian heat oscillations. These observations provided much more rapid determination of the relevant thermodynamic parameters for the step edge, even for crystals at temperatures below their roughening temperature. Use of these tools led J. Hannon and his coexperimenters to discover that some crystals behave in a highly anti-intuitive manner as their temperature is varied. The present article is devoted to a model described by a class of variational problems. The main result of the paper describes the solutions of the corresponding problem for a generic integrand.

In this paper, we study the motion planning problem for generic sub-Riemannian metrics of co-rank one. We give explicit expressions for the metric complexity (in the sense of Jean [CITE]), in terms of the elementary invariants of the problem. We construct asymptotic optimal syntheses. It turns out that among the results we show, the most complicated case is the 3-dimensional. Besides the generic C ∞ case, we study some non-generic generalizations in the analytic case.

Following the Γ-convergence approach introduced by Müller and Ortiz, the convergence of discrete dynamics for Lagrangians with quadratic behavior is established.

We consider an optimal control problem of Mayer type and prove that,under suitable conditions on the system, the value function isdifferentiable along optimal trajectories, except possibly at theendpoints. We provide counterexamples to show that this property may failto hold if some of our conditions are violated. We then apply our regularityresult to derive optimality conditions for the trajectories of the system.

We consider the eigenvalue problem $$ \begin{array}{l}\displaystyle-{\rm div} (a(|\nabla u |)\nabla u) = \lambda g(x, u) \;\mbox{in } \Omega u = 0\;\mbox{on } \partial\Omega ,\end{array}$$ in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all λ > 0 are eigenvalues.