Optimal control problems, with no discount, are studied for systems governed by nonlinear "parabolic" state equations, using a dynamic programming approach. If the dynamics are stabilizable with ...respect to cost, then the fact that the value function is a generalized viscosity solution of the associated Hamilton-Jacobi equation is proved. This yields the feedback formula. Moreover, uniqueness is obtained under suitable stability assumptions.
The regularity of the gradient of viscosity solutions of first-order Hamilton-Jacobi equations (ProQuest: Formulae and/or non-USASCII text omitted; see image) is studied under a strict convexity ...assumption on H(t,x,). Estimates on the discontinuity set of Du are derived. Such estimates imply that solutions of the above problem are smooth in the complement of a closed ^sup n^-rectifiable set. In particular, it follows that Du belongs to the classSBV, i.e., D ^sup 2^ u$ is a measure with no Cantor part.PUBLICATION ABSTRACT
The paper is concerned with an infinite-dimensional Hamilton-Jacobi equation. This equation is related to boundary control problems of Dirichlet type for semilinear parabolic systems. The viscosity ...solution approach is adapted to the equation under consideration, using the properties of fractional powers of generators of analytic semigroups. An existence-and-uniqueness result for such problem is obtained.
Under suitable controllability and smoothness assumptions, the minimum time function T(x) of a semilinear control system is proved to be locally Lipschitz continuous and semiconcave on the ...controllable set. These properties are then applied to derive optimality conditions relating optimal trajectories to the superdifferential of T.
We investigate stability properties of indirectly damped systems of evolution equations in Hilbert spaces, under new compatibility assumptions. We prove polynomial decay for the energy of solutions ...and optimize our results by interpolation techniques, obtaining a full range of power-like decay rates. In particular, we give explicit estimates with respect to the initial data. We discuss several applications to hyperbolic systems with {\em hybrid} boundary conditions, including the coupling of two wave equations subject to Dirichlet and Robin type boundary conditions, respectively.
We study the null controllability of the parabolic equation associated with the Grushin-type operator \(A=\partial_x^2+|x|^{2\gamma}\partial_y^2\,, (\gamma>0),\) in the rectangle ...\(\Omega=(-1,1)\times(0,1)\), under an additive control supported in the strip \(\omega=(a,b)\times(0,1)\,, (0<a,b<1)\). We prove that the equation is null controllable in any positive time for \(\gamma<1\), and that it fails to be so for \(\gamma>1\). In the transition regime \(\gamma=1\), we show that both behaviors live together: a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular geometric configuration, null controllability is equivalent to the observability of the Fourier components of the solution of the adjoint system uniformly with respect to the frequency.
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![[Correction]: Regularity of...](http://api.summon.serialssolutions.com/2.0.0/image/issn/XR4PS5YY4K/0022-2518/small "[Correction]: Regularity of Hamilton–Jacobi Equations when Forward Is Backward")
