In this paper we investigate the existence and Lipschitz continuity of optimal trajectories for the autonomous Bolza problem in control theory. Then, we apply our result to several examples of partial differential equations in Banach and Hilbert spaces. Recent Presentations by Control and Optimization Group Click on the highlighted titles to download the abstract, poster, slides, or other helpful information. As an application, we obtain a perimeter estimate for constrained attainable sets. This article is devoted to the study of null controllability properties for two systems of coupled one dimensional degenerate parabolic equations. The exterior measure is monotone but fails to be additive.
For this purpose, we investigate first the invariance of a bounded open domain with piecewise smooth boundary showing that such a property holds true under the same conditions that insure the invariance of the closure of the domain. Then, the natural problem that arises is how to reduce a double or multiple integral to the computation of two or more simple integrals. In this chapter, we provide a quick introduction to set-valued analysis aiming to deduce a classical theorem which guarantees the existence of a measurable selection. These results holds for systems where the control inputs are sign constrained, too. In this sense, the problem addressed in this paper can be reviewed as a switched system theory version of the classical problem of investigating whether, or to what extent, a nonlinear asymptotically controllable system admits stabilizing feedback laws. Stability by the linear approximation. In addition, the switching law is robust against time-varying and nonlinear system perturbations.
Some of these results are immediately applicable to the stability analysis of sampled data systems. Group members have produced many in the top Automation and Control Systems journals, and have presented many at leading universities. Today control theory is a rich source of basic abstract problems arising from applications, and provides an important frame of reference for investigating purely mathematical issues. For kernels that decay exponentially at infinity and possess strongly positive definite primitives, the exponential stability of weak solutions is obtained in the energy norm. This book collects some recent results, highlighting geometrical and analytical aspects and the possible connections between them.
Time-optimal problems and Fuller's phenomenon 11. . Then, we derive a simple dynamics for the propagation of singularities of general semiconcave functions. It is assumed that swimmers' bodies consist of finitely many parts, identified with the fluid they occupy, that are subsequently linked by the rotational and elastic internal forces. Nous introduisons aussi l'ensemble des points de coupure de u comme étant l'ensemble des points où aucune courbe caractéristique en temps négatif de u aboutissant en x ne peut être étendue au-delà de x en une courbe u-calibrée ; de manière équivalente, si est une courbe u-calibrée avec , alors. Furthermore, several applications are provided.
Such open loop inputs cause the system to have fairly regular solutions. Finally, under stronger assump-tionsfor the domain and Dirichlet data, we are able to conclude that solutions are globally semiconcave and semiconvex near the boundary. In optimal control,sensitivity relations are usually understood as inclusions that identify the pair formed by the dual arc and the Hamiltonian, evaluated along the associated minimizing trajectory, as a suitable generalized gradient of the value function. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Solutions can be forced to assume negative values, even when initial conditions are strictly positive. This paper studies the structure of the singular set points of nondifferentiability of viscosity solutions to Hamilton-Jacobi equations associated with general mechanical systems on the n-torus. University of Washington, 1988; Advisor: R.
For this purpose, a specific Carleman estimate for the solutions of degenerate adjoint problems is proved. Nevertheless, we are able to prove Lipschitz stability results for inverse source problems for such operators, with locally distributed measurements in arbitrary space dimension. We study a class of elliptic operators L that degenerate at the boundary of a bounded open set O subset of R-d and possess a symmetrizing invariant measure mu. Motivated by several examples coming from physics, biology, and economics, we consider a class of parabolic operators that degenerate at the boundary of the space domain. Reachable Sets and Controllability: 1. In particular, we give explicit estimates with respect to the initial data.
In this paper, we study two Energy Balance Models with Memory arising in climatology, which consist in a 1D degenerate nonlinear parabolic equation involving a memory term, and possibly a set-valued reaction term of Sellers type and of Budyko type, in the usual terminology. In such a relaxed environment the existence of solutions follows by set-valued fixed point arguments. Finally, in a standard way, we deduce null controllability also for semilinear equations. However, since its emergence, there has been a need to rethink and extend fields such as calculus of variations, differential geometry and nonsmooth analysis, which are closely tied to research on applications. Our method employs shifting the points of sign change by making use of a finite sequence of initial-value pure diffusion problems.
The traditional linear operator controllability methods based on the duality pairing do not apply to such a problem. A rather natural way to address the stabilizability problem for switched systems is to make use of periodic switching laws. As is well-known, this geometric property ensures most of the regularity results that hold for semiconcave functions, without assuming T · to be Lipschitz. Liapunov's characterization of stable matrices. Quantitative estimates of compactness, as suggested by P. Such a class is stable under linear operations, product, and pointwise convergence.
Using the distance to the closed set, we give a general necessary and sufficient condition for the invariance of K. Control theory and its applications are no exception. The result is illustrated by some simulations and examples. Such estimates proved to be extremely useful for several kinds of uniformly parabolic equations and systems. Our results apply to degenerate parabolic equations and require superlinear growth at infinity, in the gradient variables, of the Hamiltonian. We establish these estimates deriving accurate a-priori bounds on the Lipschitz, semiconcavity and semiconvexity constant of a viscosity solution when the initial data is semiconvex. The possible equilibria of such a model solve a boundary value problem for a system of nonlinear partial differential equations that we analyse when the source term is merely integrable.