We study the zero-temperature limit for Gibbs measures associated to Frenkel–Kontorova models on ${\left({ℝ}^d\right)}^{\mathbb{Z}}/{\mathbb{Z}}^d$. We prove that equilibrium states concentrate on configurations of minimal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lyapunov exponents, a bit like in the Ruelle–Pesin inequality. Then we transpose the result to certain continuous-time stationary stochastic processes associated to the viscous Hamilton–Jacobi equation. As the viscosity vanishes, the...

In this paper, the problems on purposefully controlling chaos for a three-dimensional quadratic continuous autonomous chaotic system, namely the chaotic Pehlivan-Uyaroglu system are investigated. The chaotic system, has three equilibrium points and more interestingly the equilibrium points have golden proportion values, which can generate single folded attractor. We developed an optimal control design, in order to stabilize the unstable equilibrium points of this system. Furthermore, we propose...

2000 Mathematics Subject Classification: 37F21, 70H20, 37L40, 37C40, 91G80, 93E20.In this work we will study a problem of optimal investment in financial markets with stochastic volatility with small parameter. We used the averaging method of Bogoliubov for limited development for the optimal strategies when the small parameter of the model tends to zero and the limit for the optimal strategy and demonstrated the convergence of these optimal strategies.

We consider nearly integrable, non-isochronous, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O\left(\mathrm{\mu }\right)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $T_d=O\left(\left(1/\mu \right)\log \left(1/\mu \right)\right)$ by a variational method which does not require the existence of «transition chains of tori» provided by KAM theory. We also prove that our estimate of the diffusion time $T_d$ is optimal as a consequence of a general stability result proved via classical perturbation...

The chronotherapy concept takes advantage of the circadian rhythm of cells physiology in maximising a treatment efficacy on its target while minimising its toxicity on healthy organs. The object of the present paper is to investigate mathematically and numerically optimal strategies in cancer chronotherapy. To this end a mathematical model describing the time evolution of efficiency and toxicity of an oxaliplatin anti-tumour treatment has been derived. We then applied an optimal control technique...

The chronotherapy concept takes advantage of the circadian rhythm of cells physiology in maximising a treatment efficacy on its target while minimising its toxicity on healthy organs. The object of the present paper is to investigate mathematically and numerically optimal strategies in cancer chronotherapy. To this end a mathematical model describing the time evolution of efficiency and toxicity of an oxaliplatin anti-tumour treatment has been derived. We then applied an optimal control...