Optimal feedback controllability

The classical Pontryagin maximum principle is a first-order necessary condition (FONC for short) for optimal controls of ordinary differential equations (ODEs for short). When the FONC degenerates, people introduce second-order necessary conditions (SONCs for short) for optimal controls. SONCs are well studied for optimal controls of systems described by ODEs and partial differential equations (PDEs for short). Some results for SONC of optimal controls for systems governed by stochastic differential equations (SDEs for short) are also obtained. However, nothing is known about the SONC for optimal controls of systems described by stochastic (infinite dimensional) evolution equations (SEEs for short). This paper aims to give a solution to this difficult problem, under some mild conditions.

Some second-order necessary conditions in optimal control

In recent years, the analysis and synthesis of control systems in the descriptor form have been established. This general description of dynamical systems by differential/algebraic equations is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. The present contribution deals with the design of optimal control for such dynamic systems. This means that the calculus of variations and Pontryagin's maximum principle have to be generalized for descriptor systems. An important role in the applicability of usual design algorithms plays the properness of the systems. The distinction between proper and nonproper systems is necessary in determining descriptor systems which are exclusively governed by the control inputs or additionally by their higher-order time-derivatives. In case of nonproper systems, quite different optimal control problems appear. For both cases, necessary conditions for optimal control are presented.

The structural properties of LM-g splines are investigated by optimization and optimal control theory. The continuity and structure of LM-g splines are derived by using a class of necessary conditions with state constraints of optimal control and the relationship between LM-g interpolating splines and the corresponding L-g interpolating splines. This work provides a new method for further exploration of LM-g interpolating splines and its applications in the optimal control.

In this note, we study the second order optimality conditions of optimal control problems on Riemannian manifolds. On one hand, we give the second order necessary conditions for optimal controls when the initial state is constrained to a closed subset of a Riemannian manifold. On the other hand, we obtain the second order sensitivity relations between the necessary conditions and the value function of an optimal control problem: the first and second order dual variables are related to the superjet of the value function. Both results show that the second order optimality conditions of optimal control problems on a Riemannian manifold are closely related to the curvature tensor of the manifold.

Approximate dynamic programming formulation implemented with an “adaptive critic-based” neural network structure has been shown to be a powerful technique to solve theHamilton–Jacobi–Bellman equations. As interest in this technique grows, it is important to consider the enabling factors for their possible implementations. A typical adaptive critic structure consists of two interacting neural networks; in this paper, a new architecture, called the “cost function-based single network adaptive critic” is presented that eliminates one of the networks. This approach is applicable to a wide class of nonlinear systems in engineering where the optimal control equation can be explicitly expressed in terms of the state and cost-related variables. After the “training,” the output of the neural network represents the optimal cost. Optimal control is obtained by finding the derivatives of the output of the network with respect to its input and using it in the expression for optimal control. In practical applications, there usually exist uncertainties in modeling and in system parameters. Furthermore, the controllers have operational limits. The first concern is taken care of through an “approximated system” that contains an estimate of the uncertainties from an online neural network andhelps calculate the optimal control for the changed plant. Regarding the second concern, a nonquadratic term that incorporates the control constraints is used in the performance index. Necessary conditions for optimal control are derived and an algorithm to solve the constrained-control problem with cost function-based single network adaptive critic is developed. Two aerospace systems are used to illustrate the working of the proposed technique.

Multiple shooting is an indirect solution method for complicated optimal control problems. It yields precise numerical solutions. The rate of convergence is usually quadratic due to the underlying Newton type method. However, it was recently noticed that if the general necessary conditions of optimal control are applied to state constrained problem formulations from aerospace engineering the method decelerates often to a slow linear rate of convergence. We identify the culprit, namely the appearance of a quadratic control component in the optimal control problem and certain boundary and/or interior point conditions. Finally we provide a reformulation of these boundary and/or interior point conditions in order to avoid this situation.

This study applies optimal control theory to a deterministic co-infection model of COVID-19 and tuberculosis (TB) with asymptomatic COVID-19 carriers, who are assumed to be less infectious. The optimal control strategy aims to minimize intervention costs and reduce infections by implementing five control measures, including prevention and vaccination of COVID-19, treatment of both symptomatic and asymptomatic COVID-19-infected individuals, treatment of COVID-19 and active TB co-infected individuals, and prevention of treatment failure in active TB cases. Pontryagin's minimum principle is used to characterize the necessary conditions for optimal control in reducing infections. Numerical results demonstrate the effectiveness of the optimal control strategy in suppressing diseases. The incremental cost-effectiveness ratio (ICER) for different combinations of control measures is evaluated, showing that the intervention strategy performs best when all control measures are used.

An extension of the classical Pontryagin maximum principle for Mayer problems without terminal constraints, subject to affine control systems $$\dot x = X_0 (x) + \sum\nolimits_{j = 1}^m u _j X_j (x)$$ , is proved. In connection with a suitable dilation on the state space ℝ n , we introduce a class of “homogeneous tangent vectors” which provide a nonlinear, high-order approximation of the attainable set in the case where the usual linear approximation proves to be inadequate. By studying control variations which generate homogeneous tangent vectors, we derive new necessary conditions for optimality that are particularly effective for basically nonlinear optimal control problems where other high-order tests provide no conclusive information.

We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and the density of the sources are missing and must be found along with the temperature and the free boundary. We pursue the optimal control framework analyzed in [1, 2], where the boundary heat flux, the density of the sources, and the free boundary are components of the control vector. We prove the Frechet differentiability in Besov spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov spaces for the numerical solution of the inverse Stefan problem.

In this paper, we propose a mathematical model for COVID-19-Associated Pulmonary Aspergillosis (CAPA) co-infection, that enables the study of relationship between prevention and treatment. The next generation matrix is employed to find the reproduction number. We enhanced the co-infection model by incorporating time-dependent controls as interventions based on Pontryagin's maximum principle in obtaining the necessary conditions for optimal control. Finally, we perform numerical experiments with different control groups to assess the elimination of infection. In numerical results, transmission prevention control, treatment controls, and environmental disinfection control provide the best chance of preventing the spread of diseases more rapidly than any other combination of controls.

In this paper, we consider the system governed via the coefficients of a semilinear elliptic equation and give the necessary conditions for optimal control. Furthermore, we obtain the necessary conditions for an optimal domain in a domain optimization problem.

The re-emergence of the Ebola virus disease has pushed researchers to investigate more on this highly deadly disease in order to better understand and control the outbreak and recurrence of epidemics. It is in this perspective that we formulate a realistic mathematical model for the dynamic transmission of Ebola virus disease, incorporating relevant control measures and factors such as ban on eating bush-meat, social distancing, observance of hygiene rules and containment, waning of the vaccine-induced, imperfect contact tracing and vaccine efficacy, quarantine, hospitalization and screening to fight against the spread of the disease. First, by considering the constant control parameters case, we thoroughly compute the control reproduction number [Formula: see text] from which the dynamics of the model is analyzed. The existence and stability of steady states are established under appropriate assumptions on [Formula: see text]. Also, the effect of all the control measures is investigated and the global sensitivity analysis of the control reproduction number is performed in order to determine the impact of parameters and their relative importance to disease transmission and prevalence. Second, in the time-dependent control parameters case, an optimal control problem is formulated to design optimal control strategies for eradicating the disease transmission. Using Pontryagin’s Maximum Principle, we derive necessary conditions for optimal control of the disease. The cost-effectiveness analysis of all combinations of the control measures is made by calculating the infection averted ratio and the incremental cost-effectiveness ratio. This reveals that combining the four restrictive measures conveyed through educational campaigns, screening, safe burial and the care of patients in health centers for better isolation is the most cost-effective among the strategies considered. Numerical simulations are performed to illustrate the theoretical results obtained.

In this paper we study the optimal control of an initial-boundary value problem for the classical nonviscous Cahn–Hilliard system with zero Neumann boundary conditions. Phase field systems of this type govern the evolution of diffusive phase transition processes with conserved order parameter. For such systems, optimal control problems have been studied in the past. We deal with the situation when the cost functional of the optimal control problem contains a sparsity-enhancing nondifferentiable term like the L1-norm. We establish first-order necessary and second-order sufficient optimality conditions for locally optimal controls. Since in the case without viscosity the solutions to the state and adjoint systems are considerably less regular than in the viscous case, some technical difficulties have to be overcome, and additional conditions have to be imposed. In particular, we have to restrict ourselves to the case when the nonlinearity driving the phase separation is regular, while in the presence of a viscosity term also nonlinearities of logarithmic type could be admitted. In addition, the implicit function theorem, which was employed to establish the needed differentiability properties of the control-to-state operator in the viscous case, does not apply in our situation and has to be substituted by other arguments.

This paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn--Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term in order to enhance the occurrence of sparsity effects in the optimal controls, i.e., of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions have been established for the case without sparsity, the case with sparsity has not been treated before. The results obtained in this paper also improve the known results on this phase field model for the case without sparsity.