scholarly journals Stabilization and Analytic Approximate Solutions of an Optimal Control Problem

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 476-487 ◽  
Author(s):  
Camelia Pop ◽  
Camelia Petrişor ◽  
Remus-Daniel Ene
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Equilibrium Points ◽  
Approximate Solutions ◽  
Linear Control ◽  
Runge Kutta Method ◽  
The Stability ◽  
Stability Results

Abstract This paper analyses a dynamical system derived from a left-invariant, drift-free optimal control problem on the Lie group SO(3) × ℝ3 × ℝ3 in deep connection with the important role of the Lie groups in tackling the various problems occurring in physics, mathematics, engineering and economic areas [1, 2, 3, 4, 5]. The stability results for the initial dynamics were inconclusive for a lot of equilibrium points (see [6]), so a linear control has been considered in order to stabilize the dynamics. The analytic approximate solutions of the resulting nonlinear system are established and a comparison with the numerical results obtained via the fourth-order Runge-Kutta method is achieved.

scholarly journals Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systems

2021 ◽  
Vol 31 (2) ◽  
pp. 265-284
Author(s):  
V.I. Sumin ◽  
M.I. Sumin
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Pontryagin Maximum Principle ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Lagrange Principle ◽  
The Pontryagin Maximum Principle

We consider the regularization of the classical optimality conditions (COCs) — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space $L^m_2$, the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They “overcome” the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.

Some stability results for a model of Hepatitis C including alanine aminotransferase and immune system

2020 ◽  
pp. 2050080
Author(s):  
Salvo Danilo Lombardo ◽  
Sebastiano Lombardo
Keyword(s):  
Immune System ◽  
Hepatitis C ◽  
Alanine Aminotransferase ◽  
Equilibrium Points ◽  
Asymptotical Stability ◽  
Direct Acting Antivirals ◽  
The Stability ◽  
Direct Acting ◽  
Stability Results

In clinical practice, many cirrhosis scores based on alanine aminotransferase (ALT) levels exist. Although the most recent direct acting antivirals (DAAs) reduce fibrosis and ALT levels, the Hepatitis C virus (HCV) is not always removed. In this paper, we study a mathematical model of the HCV virus, which takes into account the role of the immune system, to investigate the ALT behavior during therapy. We find five equilibrium points and analyze their stability. A sufficient condition for global asymptotical stability of the infection-free equilibrium is obtained and local asymptotical stability conditions are given for the immune-free infection and cytotoxic T lymphocytes (CTL) response equilibria. The stability of the infection equilibrium with the full immune response is numerically performed.

scholarly journals Regularized optimal control problem for a beam vibrating against an elastic foundation

2015 ◽  
Vol 63 (1) ◽  
pp. 53-71
Author(s):  
Igor Bock ◽  
Mária Kečkemétyová
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Elastic Foundation ◽  
Control Variable ◽  
Necessary Optimality Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value

Abstract We deal with an optimal control problem governed by a nonlinear hyperbolic initial-boundary value problem describing the perpendicular vibrations of a clamped beam against a u elastic foundation. A variable thickness of a beam plays the role of a control variable. The original equation for the deflection is regularized in order to derive necessary optimality conditions

scholarly journals An optimal control problem of immuno-chemotherapy in presence of gene therapy

2021 ◽  
Vol 6 (10) ◽  
pp. 11530-11549
Author(s):  
Kaushik Dehingia ◽  
◽  
Hemanta Kumar Sarmah ◽  
Kamyar Hosseini ◽  
Khadijeh Sadri ◽  
...  
Keyword(s):  
Gene Therapy ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Tumor Cells ◽  
Equilibrium Points ◽  
Time Interval ◽  
Drug Induced ◽  
Effector Cells ◽  
Proposed Model

<abstract><p>This study addresses a cancer eradication model involving effector cells in the presence of gene therapy, immunotherapy, and chemotherapy. The main objective of this study is to understand the optimal effect of immuno-chemotherpay in the presence of gene therapy. The boundedness and positiveness of the solutions in the respective feasible domains of the proposed model are verified. Conditions for which the equilibrium points of the system exist and are stable have been derived. An optimal control problem for the system has been constructed and solved to minimize the immuno-chemotherapy drug-induced toxicity to the patient. Amounts of immunotherapy to be injected into a patient for eradication of cancerous tumor cells have been found. Numerical and graphical results have been presented. From the results, it is seen that tumor cells can be eliminated in a specific time interval with the control of immuno-chemotherapeutic drug concentration.</p></abstract>

The Use of Linear Control Laws with Non-Linear Systems

1976 ◽  
Vol 9 (8) ◽  
pp. 301-306
Author(s):  
D. McLean
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Linear Systems ◽  
Linear Control ◽  
Control Laws ◽  
Optimal Regulation ◽  
Non Linear ◽  
Non Linear Systems ◽  
Linear Nature

The problem of providing optimal regulation of non-linear systems has been one of long standing and has been solved usually by empirical techniques with, seemingly, no relief provided by the methods based on the newer theories of control. This paper deals with two methods which both depend on the solution of an optimal control problem, and which take into account the specifically non-linear nature of the plant. The sub-optimal control laws which result are linear and are easily synthesised.

scholarly journals An Optimal Control Problem of Chemotherapy In Presence of Gene and Immunotherapy

2021 ◽  
Author(s):  
kaushik dehingia
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Tumor Cells ◽  
Equilibrium Points ◽  
Time Interval ◽  
Chemotherapeutic Drug ◽  
Drug Induced ◽  
Effector Cells ◽  
Proposed Model

Abstract In this study, a cancer eradication model involving effector cells in the presence of gene therapy, immunotherapy, and chemotherapy has been proposed. Boundedness and positiveness of the solutions in the respective feasible domains of the proposed model are verified. Conditions for which the equilibrium points of the system are stable have been derived. Amounts of immunotherapy to be injected to a patient for eradication of cancerous tumor cells have been found. To minimize the chemotherapy drug-induced toxicity to the patient, an optimal control problem for the system has been constructed and solved. Numerical and graphical results have been presented. Through the investigation, it was seen that tumor cells can be eliminated in a specific time interval with the control of chemotherapeutic drug concentration.

On the numerical scheme of a 2D optimal control problem with the hyperbolic system via Bernstein polynomial basis

2018 ◽  
Vol 41 (7) ◽  
pp. 1896-1903 ◽  
Author(s):  
Kamal Mamehrashi ◽  
Sohrab Ali Yousefi ◽  
Fahimeh Soltanian
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Bernstein Polynomial ◽  
Linear Time ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Polynomial Basis ◽  
Bernstein Polynomial Basis ◽  
The Given

A numerical method for solving a 2D optimal control problem (2DOCP) governed by a linear time-varying constraint is presented in this paper. The method is based upon the Bernstein polynomial basis. The properties of Bernstein polynomial functions are presented. These properties, together with the Ritz method, are then utilized to reduce the given 2DOCP to the solution of an algebraic system of equations. By solving this system, the solution of the proposed problem is achieved. The main advantage of this scheme is that the approximate solutions satisfy all initial and boundary conditions of the problem. We extensively discuss the convergence of the method. Finally, an illustrative example is included to demonstrate the validity and applicability of the new technique.

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