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Register a new userConstructive exact control of semilinear 1D heat equations
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The exact distributed controllability of the semilinear heat equation $partial_{t}y-Delta y + g(y)=f ,1_{omega}$ posed over multi-dimensional and bounded domains, assuming that $gin C^1(mathbb{R})$ satisfies the growth condition $limsup_{rto infty} g(r)/(vert rvert ln^{3/2}vert rvert)=0$ has been obtained by Fernandez-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that $g^prime$ does not grow faster than $beta ln^{3/2}vert rvert$ at infinity for $beta>0$ small enough and that $g^prime$ is uniformly Holder continuous on $mathbb{R}$ with exponent $pin [0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations.
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It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation $partial_{tt}y-partial_{xx}y + g(y)=f 1_{omega}$, with Dirichlet boundary conditions, is exactly controllable in $H^1_0(0,1)cap L^2(0,1)$ with controls $fin L^2((0,1)times(0,T))$, for any $T>0$ and any nonempty open subset $omega$ of $(0,1)$, assuming that $gin mathcal{C}^1(R)$ does not grow faster than $betavert xvert ln^{2}vert xvert$ at infinity for some $beta>0$ small enough. The proof, based on the Leray-Schauder fixed point theorem, is however not constructive. In this article, we design a constructive proof and algorithm for the exact controllability of semilinear 1D wave equations. Assuming that $g^prime$ does not grow faster than $beta ln^{2}vert xvert$ at infinity for some $beta>0$ small enough and that $g^prime$ is uniformly Holder continuous on $R$ with exponent $sin[0,1]$, we design a least-squares algorithm yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+s$ after a finite number of iterations.
A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. First- and second-order optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation of the time domains. For autonomous systems and time-independent integral cost, we prove that the Hamiltonian is constant in time when evaluated along the optimal controls and trajectories. A numerical example is provided.
The null distributed controllability of the semilinear heat equation $y_t-Delta y + g(y)=f ,1_{omega}$, assuming that $g$ satisfies the growth condition $g(s)/(vert svert log^{3/2}(1+vert svert))rightarrow 0$ as $vert svert rightarrow infty$ and that $g^primein L^infty_{loc}(mathbb{R})$ has been obtained by Fernandez-Cara and Zuazua in 2000. The proof based on a fixed point argument makes use of precise estimates of the observability constant for a linearized heat equation. It does not provide however an explicit construction of a null control. Assuming that $g^primein W^{s,infty}(mathbb{R})$ for one $sin (0,1]$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach, generalizes Newton type methods and guarantees the convergence whatever be the initial element of the sequence. In particular, after a finite number of iterations, the convergence is super linear with a rate equal to $1+s$. Numerical experiments in the one dimensional setting support our analysis.
We derive a matrix version of Li & Yau--type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R.~Hamilton did in~cite{hamilton7} for the standard heat equation. We then apply these estimates to obtain some Harnack--type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.
A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics, and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the Successive Galerkin Approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear Generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.
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