Dynamic Systems Admitting the Normal Shift and Wave Equations

We consider wave equations on Riemannian manifolds and investigate wave front dynamics in the semiclassical approximation. The problem of finding wave equations whose wave front dynamics is described by Newtonian dynamic systems admitting the normal shift is solved. A subclass of these dynamic systems that can be defined by modified Lagrange and Hamilton equations is described explicitly.     

The Cauchy problem for a class of quasilinear parabolic equations

The present paper is concerned with the Cauchy problem for the parabolic equation u_t+H(t,x,u,u)=u. New conditions guaranteeing the global classical solvability are formulated. Moreover, it is shown that the same conditions guarantee the global existence of the Lipschitz continuous viscosity solution for the related Hamilton–Jacobi equation. Mathematics Subject Classification (2000) 35K15, 35F25     

Wave Equations in Riemannian Spaces

With regard to applications in quantum theory, we consider the classical wave equation involving the scalar curvature with an arbitrary coefficient . General properties of this equation and its solutions are studied based on modern results in group analysis with the aim to fix a physically justified value of . These properties depend essentially not only on the values of and the mass parameter but also on the type and dimension of the space. Form invariance and conformal invariance must be distinguished in general. A class of Lorentz spaces in which the massless equation satisfies the Huygens principle and its Green's function is free of a logarithmic singularity exists only for the conformal value of . The same value of follows from other arguments and the relation to the known WKB transformation method that we establish.     

Separation of Variables in the Kovalevskaya–Goryachev–Chaplygin Gyrostat

We construct separation variables for the Kovalevskaya–Goryachev–Chaplygin gyrostat for arbitrary values of the parameters. We show that different separation variables can be constructed for the same integrable system if different integrals of motion are chosen.     

Function Spaces as Path Spaces of Feller Processes

Let {X_t}_{t ≥ 0} be a Feller process with infinitesimal generator (A, D(A)). If the test functions are contained in D(A), —A |C^∞_c (ℝⁿ) is a pseudo–differential operator p(x, D) withsymbol p(x, ξ). We investigate local and global regularity properties of the sample paths t ↦ X_t in terms of (weighted) Besov B^s_pq (ℝ, ρ) and Triebel–Lizorkin F^s_pq (ℝ, ρ) spaces. The parameters for these spaces are determined by certain indices that describe the asymptotic behaviour of the symbol p(x, ξ). Our results improve previous papers on Lévy [5, 9] and Feller processes [22].     

Optimal Control of Rigid Body Angular Velocity with Quadratic Cost

In this paper, we consider the problem of obtaining optimal controllers which minimize a quadratic cost function for the rotational motion of a rigid body. We are not concerned with the attitude of the body and consider only the evolution of the angular velocity as described by the Euler equations. We obtain conditions which guarantee the existence of linear stabilizing optimal and suboptimal controllers. These controllers have a very simple structure.     

Toda Chains in the Jacobi Method

We use the Jacobi method to construct various integrable systems, such as the Stäckel systems and Toda chains, related to various root systems. We find canonical transformations that relate integrals of motion for the generalized open Toda chains of types B _n, C _n, and D _n.     

Feedback Control Methodologies for Nonlinear Systems

A number of computational methods have been proposed in the literature to design and synthesize feedback controls when the plant is modeled by nonlinear dynamics. However, it is not immediately clear which is the best method for a given problem; this may depend on the nature of the nonlinearities, size of the system, whether the amount of control used or time needed for the method is a concern, and other factors. In this paper, a comprehensive comparison study of five methods for the synthesis of nonlinear control systems is carried out. The performance of the methods on several test problems are studied, and some recommendations are made as to which feedback control method is best to use under various conditions.     

Large time behavior of solutions of Hamilton–Jacobi equations with periodic boundary data

We study the large time behavior of viscosity solutions of Hamilton–Jacobi equations with periodic boundary data on bounded domains. We establish a result on convergence of viscosity solutions to state constraint asymptotic solutions or periodic asymptotic solutions depending on the sign of critical value as time goes to infinity.     

The evolutionary limit for models of populations interacting competitively via several resources

We consider an integro-differential nonlinear model that describes the evolution of a population structured by a quantitative trait. The interactions between individuals occur by way of competition for resources whose concentrations depend on the current state of the population. Following the formalism of Diekmann et al. (2005) [16], we study a concentration phenomenon arising in the limit of strong selection and small mutations. We prove that the population density converges to a sum of Dirac masses characterized by the solution φ of a Hamilton-Jacobi equation which depends on resource concentrations that we fully characterize in terms of the function φ.     

Small noise asymptotics for invariant densities for a class of diffusions: A control theoretic view

We consider multi-dimensional nondegenerate diffusions with invariant densities, with the diffusion matrix scaled by a small >0. The o.d.e. limit corresponding to =0 is assumed to have the origin as its unique globally asymptotically stable equilibrium. Using control theoretic methods, we show that in the ↓0 limit, the invariant density has the form ≈exp(−W(x)/²), where the W is characterized as the optimal cost of a deterministic control problem. This generalizes an earlier work of Sheu. Extension to multiple equilibria is also given.     

关于AKNS族和一个耦合的MKdV族的规范等价可积系统与r-矩阵

得到了一个新的耦合的MKdV族．通过规范变换，首次从AKNS族中得到耦合的MKdV族的Lax表示、可积系与约束流；利用Lax表示，构造了耦合MKdV族的约束流的γ-矩阵，同时也给出了该方程族约束流的第二组守恒积分与对合性．     

Multi-Target Control Problems

We study control problems with several targets in the case of nonlinear dynamic systems. The map associating with every initial condition the minimal time to reach successively two given targets is characterized in the framework of differential inclusions through the notion of viability kernel. This approach allows one to treat the problem without assumptions of regularity and to build numerical schemes computing the minimal time. We also study the problem where an order of visit of the targets is required. The statements are also extended to the case of p targets under state constraints. Equivalent formulations in terms of Hamilton–Jacobi equations are also provided.     

A double obstacle problem arising in differential game theory

A Hamilton–Jacobi equation involving a double obstacle problem is investigated. The link between this equation and the notion of dual solutions—introduced in [S. As Soulaimani, Infinite horizon differential games with asymmetric information, PhD thesis; P. Cardaliaguet, Differential games with asymmetric information, SIAM J. Control Optim. 46 (3) (2007) 816–838; P. Cardaliaguet, C. Rainer, Stochastic differential games with asymmetric information, Appl. Math. Optim. 59 (1) (2009) 1–36] in the framework of differential games with lack of information—is established. As an application we characterize the convex hull of a function in the simplex as the unique solution of some nonlinear obstacle problem.     

Large deviations estimates for some non-local equations: Fast decaying kernels and explicit bounds

We study large deviations for some non-local parabolic type equations. We show that, under some assumptions on the non-local term, problems defined in a bounded domain converge with an exponential rate to the solution of the problem defined in the whole space. We compute this rate in different examples, with different kernels defining the non-local term, and it turns out that the estimate of convergence depends strongly on the decay at infinity of that kernel.     

Guaranteed Output Feedback Control for Uncertain Systems under Control and State Constraints

This paper deals with output feedback control of uncertain nonlinear dynamic systems in the presence of state and control constraints. We provide necessary and sufficient conditions for the guaranteed viability property defined by the existence of a Lipschitz closed-loop that maintains exactly the state of the system in a given closed domain of constraints despite some bounded disturbances acting both on the dynamics and on the output. Using the notion of Lipschitz kernel of a closed set-valued map, we obtain some equivalent geometric and Hamilton–Jacobi–Isaac conditions for the guaranteed viability property. We derive an algorithm to build a guaranteed viable output feedback.