Numerical solution to the optimal birth feedback control of a population dynamics: viscosity solution approach

Abstract

SUMMARYThis paper is concerned with the optimal birth control of a McKendrick-type age-structured population dynamic system. We use the dynamic programming approach in our investigation. The Hamilton-JacobiBellman equation satisfied by the value function is derived. It is shown that the value function is the viscosity solution of the Hamilton-Jacobi-Bellman equation. The optimal birth feedback control is found explicitly through the value function. A finite difference scheme is designed to obtain the numerical solution of the optimal birth feedback control. The validity of the optimality of the obtained control is verified numerically by comparing with different controls under the same constraint. All the data utilized in the computation are taken from the census of the population of China in 1989.

Optimal Propagating Fronts Using Hamilton-Jacobi Equations

“…where τ > 0 modulates the smoothness of the approximation. The solution of (6) with (7) was found by the approach presented in [27] (see also [30]), where the minimization in (7) was obtained by using the fmincon routine of the Matlab Optimization Toolbox.…”

“…Similar conditions are proved for the extension of the same problem to an infinite horizon. Since analytic solutions of such problems are almost impossible to find, approximate solutions are searched for by using a numerical scheme based on the method proposed in [27]. The resulting approach is compared with that of [24,28] both in terms of computational time and ability to optimize the cost functional, thus highlighting the effectiveness of the proposed technique.…”

Adaptive boundary stabilization for first‐order hyperbolic PDEs with unknown spatially varying parameter

“…From (3) and (5), we see that k.x; y; t/ D N k.x y; t/ for some continuous function N k. /. Then, there holds Taking inverse Laplace transform in s to both sides of the previous equation yields (6).…”

“…In the past few years, much effort has been made on the control of first-order hyperbolic PDEs, because these PDEs can be used to model many physical phenomenons, such as chemical reactors, heat exchangers, traffic flows, and population dynamics (e.g., [1][2][3][4][5][6][7][8][9][10][11][12] and references therein). Specifically, by semigroup theory, Guo and Sun [6] and Sano [7] investigated the optimal control and the output tracking for first-order linear hyperbolic PDEs, respectively. In [8], the infinite-dimensional backstepping method was successfully extended from parabolic PDEs to first-order hyperbolic PDEs to achieve boundary stabilization, and subsequently, in [9][10][11], the case with known spatially varying parameter was solved.…”

A Stochastic Model for Population and Well-Being Dynamics

“…Farkas (2004) focuses on the stability of nonlinear McKendrick equations with fertility and mortality rates. Finally, Guo and Sun (2005) study the optimization of birth with a structured McKendrick model by using a dynamic-programming-based method of approximation to study the case of population in China. In this model, the predictions for the 1989-2014 period are compared with an objective that they call ''ideal population.…”