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Boundary and distributed optimal control for a population dynamics PDE model with discontinuous in time Galerkin FEM schemes

EFthymios N. Karatzas

Abstract

We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model is approximated by conforming finite element methods mimicking the spatial part, while a discontinuous Galerkin method is used for the time discretization. We investigate the sensitivity of the solution distance from the target function, in cases with smooth and rough initial data. We employ low, and higher-order polynomials in time and space whenever proper regularity is present. The approximation schemes considered are with and without control constraints, driving efficiently the system to desired states realized using non-linear gradient methods.

Boundary and distributed optimal control for a population dynamics PDE model with discontinuous in time Galerkin FEM schemes

Abstract

We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model is approximated by conforming finite element methods mimicking the spatial part, while a discontinuous Galerkin method is used for the time discretization. We investigate the sensitivity of the solution distance from the target function, in cases with smooth and rough initial data. We employ low, and higher-order polynomials in time and space whenever proper regularity is present. The approximation schemes considered are with and without control constraints, driving efficiently the system to desired states realized using non-linear gradient methods.
Paper Structure (25 sections, 12 theorems, 69 equations, 4 figures, 5 tables, 1 algorithm)

This paper contains 25 sections, 12 theorems, 69 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The physical model and related results
  3. Background
  4. Notation
  5. Preliminaries and discretization tools
  6. Distributed control, Dirichlet zero boundary and control constraints.
  7. The continuous and discrete control problem/optimality system
  8. The fully-discrete distributed optimal control problem
  9. The discrete distributed optimality system
  10. Second order optimality conditions
  11. Cones of Critical Directions
  12. Sufficient conditions for optimality
  13. Robin boundary control
  14. The optimality system
  15. The fully discrete Robin boundary type optimal control problem
  16. ...and 10 more sections

Key Result

Lemma 3.2

The cost functional $J:L^2[0,T;{L^2(\Omega)}] \to \mathbb R$ is of class $C^{\infty}$ and for every $g_1,g_2,y_1,y_2 \in L^2[0,T;{L^2(\Omega)}]$, where $\mu _i({\color{black}g_1,g_2}) \equiv {\mu}_{g_i} \in W_D(0,T), i=1,2,$ is the unique solution of following problem: For all $v \in L^2[0,T;H^1(\Omega)] \cap H^1[0,T;H^{-1}(\Omega)]$, where $\mu_{g_i}(T)=0$ and $(\mu_{g_i})_t \in L^2[0,T;H^{-1}(

Figures (4)

  • Figure 5.1: The $y_1(0,x_1,x_2)$ smooth initial data, while $y_2(0,x_1,x_2)=25$ everywhere in $\Omega$.
  • Figure 5.2: The $y_1(0,x_1,x_2)$ and $y_2(0,x_1,x_2)$ rough initial data.
  • Figure 5.3: Phaseplanes for the $y_1(t,x,y)$ and $y_2(t,x,y)$ applying control $(g_1,g_2)=(0,0)$, $(0.1,0.1)$, $(1,1)$, $(1.5,1.5)$, $(2.5,2.5)$, $(4,4)$ and for parameters values: prey diffusion $\epsilon_1=0.1$, predator diffusion $\epsilon_2=0.01$, growth rate of prey $a=0.47$, searching efficiency/attack rate $b=0.024$, predator's attack rate and efficiency at turning food into offspring-conversion efficiency $c=0.023$, predator mortality rate $d=0.76$.
  • Figure 5.4: Nullclines for the $y_1(t,x,y)$ and $y_2(t,x,y)$ applying controls $(g_1,g_2)=(0,0)$, $(1,1)$, $(2,2)$, $(4,4)$ and for parameters: prey diffusion $\epsilon_1=0.1$, predator diffusion $\epsilon_2=0.01$, the growth rate of prey $a=0.47$, searching efficiency/attack rate $b=0.024$, predator's attack rate and efficiency on turning food into offspring-conversion efficiency $c=0.023$, predator mortality rate $d=0.76$.

Theorems & Definitions (22)

  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Definition 3.5
  • Lemma 3.6
  • Lemma 3.7
  • proof
  • Theorem 3.8
  • ...and 12 more