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Nous nous proposons, dans ce travail, d'étudier certaines propriétés géométriques telles que diverses symétries et diverses concavités radiales, directionnelles, etc., pour des équations completement non linéaires (...).

The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form: ${\displaystyle \underset{{\displaystyle (u,v)\in {𝒰}_{ad}}}{inf}{\int }_0^{1}f\left(t,u\left({\theta }_v\left(t\right)\right),u^{\text{'}}\left(t\right),v\left(t\right)\right)\mathrm{d}t}$, (1) where ${𝒰}_{ad}$ is a set of admissible controls and ${\theta }_v$ is the solution of the following equation: $\{\frac{\mathrm{d}\theta \left(t\right)}{\mathrm{d}t}=g(t,\theta \left(t\right),v\left(t\right)),t\in [0,1]$ ; ${\displaystyle \theta \left(0\right)={\theta }_0,\theta \left(t\right)\in [0,1]\forall t}$. (2). The results are nonlocal and new.

The aim of this paper is to study problems of the form: $in{f}_{(u\in V)}J\left(u\right)$ with $J\left(u\right):={\int }_0^{1}L(s,y_u\left(s\right),u\left(s\right))\mathrm{d}s+g\left(y_u\left(1\right)\right)$ where is a set of admissible controls and is the solution of the Cauchy problem: $\dot{x}\left(t\right)=\langle f(.,x(.)),{\nu }_t\rangle +u\left(t\right),t\in (0,1)$, $x\left(0\right)=x_0$ and each ${\nu }_t$ is a nonnegative measure with support in . After studying the Cauchy problem, we establish existence of minimizers, optimality conditions (in particular in the form of a nonlocal version of the Pontryagin principle) and prove some regularity results. We also consider the more general case where the control also enters...

This article is devoted to the optimal control of state equations with memory of the form: $\dot{x}\left(t\right)=F(x\left(t\right),u\left(t\right),{\int }_0^{+\infty }A\left(s\right)x(t-s)\mathrm{d}s),t>0,$ with initial conditions $x\left(0\right)=x,x(-s)=z\left(s\right),s>0$. Denoting by $y_{x,z,u}$ the solution of the previous Cauchy problem and: $v(x,z):={inf}_{u\in V}\left\{{\int }_0^{+\infty }{\mathrm{e}}^{-\lambda s}L(y_{x,z,u}\left(s\right),u\left(s\right))\mathrm{d}s\right\}$ 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: ...