Optimality Conditions and Synthesis for the Minimum Time Problem

We study the minimum time optimal control problem for a nonlinear system in R ⁿ with a general target. Necessary and sufficient optimality conditions are obtained. In particular, we describe a class of costates that are included in the superdifferential of the minimum time function, even in the case when this function is only lower semicontinuous. Two set-valued maps are constructed to provide time optimal synthesis.     

Cosmically Lipschitz Set-Valued Mappings

This paper introduces a kind of sub-Lipschitz continuity for set-valued mappings based on the cosmic metric. This type of Lipschitz behavior has applications with regards to necessary optimality conditions, the Hamilton–Jacobi equation, and invariance of unbounded differential inclusions. Cosmically Lipschitz assumptions allow for broader applications than previously allowed under Lipschitz assumptions. It is also shown that a cosmically Lipschitz mapping can be characterized by the normal cones to its graph using the coderivative, and various rules are presented in order to more easily identify such a mapping.     

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     

Regular Solutions of First-Order Hamilton–Jacobi Equations for Boundary Control Problems and Applications to Economics

This is the first of two papers regarding a family of linear convex control problems in Hilbert spaces and the related Hamilton–Jacobi–Bellman equations. The framework is motivated by an application to boundary control of a PDE modeling investments with vintage capital. Existence and uniqueness of a strong solution (namely, the limit of classic solutions of approximating equations, introduced by Barbu and Da Prato) is investigated. Moreover, such a solution is proved to be C¹ in the space variable.     

Scale of Viability and Minimal Time of Crisis

In this paper, we introduce and study the minimal time of a crisis map which measures the minimal time spent outside a given closed domain of constraints by trajectory solutions of a differential inclusion. The interest of such a notion is basically to tackle simultaneously viability and target issues. The main mathematical result characterizes the epigraph of the crisis map in terms of a viability kernel of an augmented problem. This allows the obtaining of the numerical schemes we specify and to derive an equivalent Hamilton–Jacobi formulation. A simple economic example illustrates the results.     

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.     

变捕捞努力量收获模型控制

利用微分包含给出了努力量可变的收获模型,基于生存理论和求解线性规划给出了将单种群数量控制在某范围的方法.最后对于常用的Logistic模型,证明只要控制努力量就可以将种群数量控制在指定范围内.     

On the Dynamics of a Differential Inclusion Built upon a Nonconvex Constrained Minimization Problem

In this paper, we study the dynamics of a differential inclusion built upon a nonsmooth, not necessarily convex, constrained minimization problem in finite-dimensional spaces. In particular, we are interested in the investigation of the asymptotic behavior of the trajectories of the dynamical system represented by the differential inclusion. Under suitable assumptions on the constraint set and the two involved functions (one defining the constraint set, the other representing the functional to be minimized), it is proved that all the trajectories converge to the set of the constrained critical points. We present also a large class of constraint sets satisfying our assumptions. As a simple consequence, in the case of a smooth convex minimization problem, we have that any trajectory converges to the set of minimizers.Research partially supported by the research project CNR-GNAMPA Mathematical Methods for Control Theory and supported in part by the European Communitys Human Potential Program under Contract HPRN-CT-2002-00281, Evolution Equations.     

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.     

Lipschitz Continuity of the Value Function in Optimal Control

For optimal control problems in with given target and free final time, we obtain a necessary and sufficient condition for local Lipschitz continuity of the optimal value as a function of the initial position. The target can be an arbitrary closed set, and the dynamics can depend in a measurable way on the time. As a limit case of this condition, we obtain a characterization of the viability property of the target, in terms of perpendiculars to the target instead of tangent cones. As an application, we analyze the convergence of certain discretization schemes for time-optimal problems.     

History Path Dependent Optimal Control and Portfolio Valuation and Management

Regarding the evolution of financial asset prices governed by an history dependent (path dependent) dynamical system as a prediction mechanism, we provide in this paper the dynamical valuation and management of a portfolio (replicating for instance European, American and other options) depending upon this prediction mechanism (instead of an uncertain evolution of prices, stochastic or tychastic). The problem is actually set in the format of a viability/capturability theory for history dependent control systems and some of their results are then transferred to the specific examples arising in mathematical finance or optimal control. They allow us to provide an explicit formula of the valuation function and to show that it is the solution of a ``Clio Hamilton–Jacobi–Bellman' equation. For that purpose, we introduce the concept of Clio derivatives of ``history functionals' in such a way we can give a meaning to such an equation. We then obtain the regulation law governing the evolution of optimal portfolios.     

Error Estimates for a Class of Elliptic Optimal Control Problems

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.     

Projectively Equivalent Riemannian Spaces as Quasi-bi-Hamiltonian Systems

The class of Riemannian spaces admitting projectively, or geodesically, equivalent metrics is very closely related to a certain class of spaces for which the Hamilton–Jacobi equation for geodesics is separable. This fact is established, and its consequences explored, by showing that when a Riemannian space has a projectively equivalent metric its geodesic flow is a quasi-bi-Hamiltonian system. The existence of involutive first integrals of the geodesic flow, quadratic in the momenta, follows by a standard type of argument. When these integrals are independent they generate a Stäckel system.     

Nonlinear Impulse Target Problems under State Constraint: A Numerical Analysis Based on Viability Theory

We study the problem of reaching a target without leaving a prescribed set for controlled impulse dynamics. First, we provide a numerical procedure for the approximation of the set of initial conditions from which the objective can be met. Then we show that the minimum time function associated with this target problem can be approached by a sequence of value functions for suitable discrete-time systems. This can be deduced from the fact that the epigraph of the minimum time function is the set of initial conditions from which a target can be reached without leaving a constraint set for an auxiliary impulse system. In this case, the numerical procedure for the qualitative target problem can be simplified. We provide results for estimating the convergence rate of the simplified scheme.     

Weak KAM theorem for Hamilton‐Jacobi equations with Neumann boundary conditions on noncompact manifolds

In this paper, we consider Hamilton–Jacobi equations with homogeneous Neumann boundary condition. We establish some results on noncompact manifold with homogeneous Neumann boundary conditions in view of weak Kolmogorov‐Arnold‐Moser (KAM) theory, which is a generalization of the results obtained by Fathi under the non‐bounded condition. Copyright © 2014 John Wiley & Sons, Ltd.     

Macrocosmos/microcosmos: celestial mechanics and old quantum theory

The Bohr atom was a solar system in miniature. Despite many deep foundational questions related to the origin of quantized motion, rapid progress was made in its mathematical development and its apparently successful application to spectral line series. In United States, where celestial mechanics flourished throughout the 19th and well into the 20th century, mathematicians and physicists were well prepared for just this sort of problem and made it their own far faster than many areas of the new physics. This paper examines the link between classical problems of perturbation theory, three-body and N-body orbital trajectories, the Hamilton–Jacobi equation, and the old quantum theory. I discuss why it was comparatively easy for American applied mathematicians, astronomers, and mathematical physicists to make significant contributions quickly to quantum theory and why further progress toward quantum mechanics by the same cohort was, in contrast, so slow.     

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.     

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.     

A Geometric Characterization of Viable Sets for Controlled Degenerate Diffusions

For a general controlled diffusion process and an arbitrary closed set K we study the viability, or weak invariance, or controlled invariance, of K, that is, the existence of a control for each initial point in K keeping the trajectory forever in K. By viscosity solutions methods we prove a simple necessary and sufficient condition involving only a deterministic second-order normal cone to K and the data of the diffusion process. We also give an extension to stochastic differential games.     

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.