This article is devoted to the optimal control of state equations with memory of the form:

$\dot{x}(t)=F(x(t),u(t), \int_0^{+\infty} A(s) x(t-s) {\rm d}s), \; t>0,$

with initial conditions $x(0)=x, \; x(-s)=z(s), s>0$ .

Denoting by $y_{x, z, u}$ the solution of the previous Cauchy problem and:

$v(x,z):=\inf_{u\in V} \lbrace \int_0^{+\infty} {\rm e}^{-\lambda s } L(y_{x,z,u}(s), u(s)){\rm d}s\rbrace$

where V is a class of admissible controls, we prove that v is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form:

$\lambda v(x,z)+H(x,z,\nabla_x v(x,z))+\langle D_z v(x,z), \dot{z} \rangle=0$

in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.

The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a ersatz material approximation [Allaire et al., J. Comput. Phys.194 (2004) 363–393], a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limitations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic error committed by using the ersatz material approximation and, on a model case, explain that they amplifies instabilities by a second order analysis of the objective function.

In this paper we construct upper bounds for families offunctionals of the form

$$E_\varepsilon(\phi):=\int_\Omega\Big(\varepsilon |\nabla\phi|^2+\frac{1}{\varepsilon }W(\phi)\Big){\rm d}x+\frac{1}{\varepsilon }\int_{{\mathbb{R}}^N}|\nabla \bar H_{F(\phi)}|^2{\rm d}x$$

where Δ $\bar H_u$ = div { $\chi_\Omega$ u}. Particular cases of such functionals arise inMicromagnetics. We also use our technique to construct upper boundsfor functionals that appear in a variational formulation ofthe method of vanishing viscosity for conservation laws.

We consider some discrete and continuous dynamics in a Banach spaceinvolving a non expansive operator J and a corresponding family ofstrictly contracting operators Φ (λ, x): = λJ( $\frac{1-\lambda}{\lambda}$ x) for λ ∈ ] 0,1] . Our motivationcomes from the study of two-player zero-sum repeated games, wherethe value of the n-stage game (resp. the value of theλ-discounted game) satisfies the relationv n = Φ( $\frac{1}{n}$ , $v_{n-1}$ ) (resp.  $v_\lambda$ = Φ(λ, $v_\lambda$ )) where J is the Shapleyoperator of the game. We study the evolution equationu'(t) = J(u(t))- u(t) as well as associated Eulerian schemes,establishing a new exponential formula and a Kobayashi-likeinequality for such trajectories. We prove that the solution of thenon-autonomous evolution equationu'(t) = Φ(λ(t), u(t))- u(t) has the same asymptoticbehavior (even when it diverges) as the sequence v n (resp. as thefamily $v_\lambda$ ) when λ(t) = 1/t (resp. whenλ(t) converges slowly enough to 0).

In this paper, we study the stabilization of a two-dimensional Burgers equation around a stationary solution by a nonlinear feedback boundary control. We are interested in Dirichlet and Neumann boundary controls. In the literature, it has already been shown that a linear control law, determined by stabilizing the linearized equation, locally stabilizes the two-dimensional Burgers equation. In this paper, we define a nonlinear control law which also provides a local exponential stabilization of the two-dimensional Burgers equation. We end this paper with a few numerical simulations, comparing the performance of the nonlinear law with the linear one.

In this paper, we consider a linear program with only equality constraints but containing interval and random coefficients. We first address the linear program with interval coefficients, and establish some structural properties of this linear program. On this basis, a solution method is proposed. We then move on to consider the linear program with random coefficients. Using the chance constraint approach and a new approach, the satisfaction degree approach, we obtain the two respective deterministic equivalent formulations. Then the results and the numerical solution methods obtained for these two linear models are applied to the original linear problem which contains both interval and random coefficients. By way of illustration, we consider a practical problem, where the optimal mixing proportions need to be determined for the mix slurry in the production process of aluminium with sintering. This gives rise to a linear program with interval and random coefficients. Its deterministic equivalent formulations are presented. Preliminary numerical examples show that the proposed models and the solution methods are promising.

In boats used for competitive rowing it is traditional for the rowers to use strokes in which the angle between the oar shaft and the perpendicular to the hull centre line is much greater at the catch than it is at the end of the power stroke. As a result, the oar blade is even more inefficient in its action at the catch than it is at the end of the power stroke. This paper shows how boat performance in a race would be improved by reducing the difference in these starting and finishing angles.The claim of improved race performance is supported by a detailed investigation of the dynamics involved in the case of a particular coxless pair whose performance has been recorded by the Australian Institute of Sport. We also suggest an easy way to make the necessary change in boat design.

A class of first-order impulsive functional differential equations with forcing terms is considered. It is shown that, under certain assumptions, there exist positive T-periodic solutions, and under some other assumptions, there exists no positive T-periodic solution. Applications and examples are given to illustrate the main results.