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Register a new userHybrid Optimal Control Problems for a Class of Semilinear Parabolic Equations
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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.
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Stabilizing feedback operators are presented which depend only on the orthogonal projection of the state onto the finite-dimensional control space. A class of monotone feedback operators mapping the finite-dimensional control space into itself is considered. The special case of the scaled identity operator is included. Conditions are given on the set of actuators and on the magnitude of the monotonicity, which guarantee the semiglobal stabilizing property of the feedback for a class semilinear parabolic-like equations. Subsequently an optimal feedback control minimizing the quadratic energy cost is computed by a deep neural network, exploiting the fact that the feedback depends only on a finite dimensional component of the state. Numerical simulations demonstrate the stabilizing performance of explicitly scaled orthogonal projection feedbacks, and of deep neural network feedbacks.
An abstract framework guaranteeing the local continuous differentiability of the value function associated with optimal stabilization problems subject to abstract semilinear parabolic equations subject to a norm constraint on the controls is established. It guarantees that the value function satisfies the associated Hamilton-Jacobi-Bellman equation in the classical sense. The applicability of the developed framework is demonstrated for specific semilinear parabolic equations.
In PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of a (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the POD model is usually expensive, however, and therefore often impossible in a model-predictive control (MPC) context. Thus, reduced models of mediocre quality might be accepted. We take the view of a simplified Newton method for solving semilinear evolution equations to derive an algorithm that can serve as an offline phase to produce a POD model. Approaches that build the POD model with impulse response snapshots can be regarded as the first Newton step in this context. In particular, POD models that are based on impulse response snapshots are extended by adding a second simplified Newton step. This procedure improves the approximation quality of the POD model significantly by introducing a moderate amount of extra computational costs during optimization or the MPC loop. We illustrate our findings with an example satisfying our assumptions.
We show a triviality result for pointwise monotone in time, bounded eternal solutions of the semilinear heat equation begin{equation*} u_{t}=Delta u + |u|^{p} end{equation*} on complete Riemannian manifolds of dimension $n geq 5$ with nonnegative Ricci tensor, when $p$ is smaller than the critical Sobolev exponent $frac{n+2}{n-2}$.
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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