Optimal control of harvested population at the edge of extinction in an unprotected area, is considered. The underlying population dynamics is governed by a Kolmogorov-Petrovsky-Piskunov equation with a harvesting term and space-dependent coefficients while the control consists of transporting individuals from a natural reserve. The nonlinear optimal control problem is approximated by means of a Galerkin scheme. Convergence result about the optimal controlled solutions and error estimates between the corresponding optimal controls, are derived. For certain parameter regimes, nearly optimal solutions are calculated from a simple logistic ordinary differential equation (ODE) with a harvesting term, obtained as a Galerkin approximation of the original partial differential equation (PDE) model. A critical allowable fraction of the reserve's population is inferred from the reduced logistic ODE with a harvesting term. This estimate obtained from the reduced model allows us to distinguish sharply between survival and extinction for the full PDE itself, and thus to declare whether a control strategy leads to success or failure for the corresponding rescue operation while ensuring survival in the reserve's population. In dynamical terms, this result illustrates that although continuous dependence on the forcing may hold on finite-time intervals, a high sensitivity in the system's response may occur in the asymptotic time. We believe that this work, by its generality, establishes bridges interesting to explore between optimal control problems of ODEs with a harvesting term and their PDE counterpart.
Optimal control problems of nonlinear delay equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.
Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated here in terms of optimal control of energy balance climate models posed on the sphere S2.
The goal of this article is to propose an efficient way of empirically improving suboptimal solutions designed from the recent method of finite-horizon parameterizing manifolds (PMs) introduced by Chekroun and Liu (Acta Appl. Math., 2015) and concerned with the (sub)optimal control of nonlinear parabolic partial differential equations (PDEs). Given a finite horizon [0, T ] and a reduced low-mode phase space, a finite-horizon PM provides an approximate parameterization of the high modes by the low ones so that the unexplained high-mode energy is reduced — in an L 2-sense — when this parameterization is applied. In Chekroun and Liu (Acta Appl. Math., 2015), various PMs were constructed analytically from the uncontrolled version of the underlying PDE that allow for the design of reduced systems from which low-dimensional suboptimal controllers can be efficiently synthesized. In this article, the analytic approach from Chekroun and Liu (Acta Appl. Math., 2015) is recalled and a post-processing procedure is introduced to improve the PM-based suboptimal controllers. It consists of seeking for a high-mode parametrization aiming to reduce the energy contained in the high modes of the PDE solution, when the latter is driven by a PM-based suboptimal controller. This is achieved by solving simple regression problems. The skills of the resulting empirically post-processed suboptimal controllers are numerically assessed for an optimal control problem associated with a Burgers-type equation.
This article proposes a new approach based on finite-horizon parameterizing manifolds (PMs) for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. Given a finite horizon [0,T] and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the uncontrolled high modes by the controlled low ones so that the unexplained high-mode energy is reduced, in an L2-sense, when this parameterization is applied. Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes. These formulas allow for an effective derivation of reduced ODE systems, aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. A priori error estimates between the resulting PM-based low-dimensional suboptimal controller u_R* and the optimal controller u* are derived. These estimates demonstrate that the closeness of u_R* to u*? is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with u_R* and u*; and (ii) the energy kept in the high modes of the PDE solution either driven by u_R* or u* itself. The practical performances of such PM-based suboptimal controllers are numerically assessed for various optimal control problems associated with a Burgers-type equation. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results. The practical performances of such PM-based suboptimal controllers are numerically assessed for optimal control problems associated with a Burgers-type equation; the locally as well as globally distributed cases being both considered. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.