Abstract theory of optimal control. IV

1980 ◽  
Vol 20 (5) ◽  
pp. 800-821
Author(s):  
V. A. Yakubovich
Keyword(s):  
Optimal Control ◽  
Abstract Theory ◽  
Theory Of Optimal Control

Stability Analysis of a Mathematical Model for Glioma-Immune Interaction under Optimal Therapy

2019 ◽  
Vol 20 (3-4) ◽  
pp. 269-285 ◽  
Author(s):  
Subhas Khajanchi
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Stability Analysis ◽  
Model Simulation ◽  
Equilibrium Points ◽  
Glioma Cells ◽  
Adjoint System ◽  
Cellular Immunotherapy ◽  
Theory Of Optimal Control ◽  
Malignant Glioma Cells

AbstractWe investigate a mathematical model using a system of coupled ordinary differential equations, which describes the interplay of malignant glioma cells, macrophages, glioma specific CD8+T cells and the immunotherapeutic drug Adoptive Cellular Immunotherapy (ACI). To better understand under what circumstances the glioma cells can be eliminated, we employ the theory of optimal control. We investigate the dynamics of the system by observing biologically feasible equilibrium points and their stability analysis before administration of the external therapy ACI. We solve an optimal control problem with an objective functional which minimizes the glioma cell burden as well as the side effects of the treatment. We characterize our optimal control in terms of the solutions to the optimality system, in which the state system coupled with the adjoint system. Our model simulation demonstrates that the strength of treatment $u_{1}(t)$ plays an important role to eliminate the glioma cells. Finally, we derive an optimal treatment strategy and then solve it numerically.

scholarly journals Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

2019 ◽  
Vol 25 ◽  
pp. 15 ◽  
Author(s):  
Manh Khang Dao
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Recent Work ◽  
Comparison Principle ◽  
Value Function ◽  
Exit Costs ◽  
Theory Of Optimal Control ◽  
New Feature ◽  
Jacobi Equations ◽  
The Value Function

We consider an optimal control on networks in the spirit of the works of Achdou et al. [NoDEA Nonlinear Differ. Equ. Appl. 20 (2013) 413–445] and Imbert et al. [ESAIM: COCV 19 (2013) 129–166]. The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. [ESAIM: COCV 21 (2015) 876–899] and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016) 535–545].

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