Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

In this study, the problem of finding an optimal controller for nonlinear systems with one input and a reference tracking signal is approached. With the problem's formulation, any desired signal can be tracked instantly with a closed-loop controller without the need for integral terms. Presentation lies at the heart of optimal control. This study, however, does not consider the integral term, allowing tracking and stability to occur naturally. It has a broad scope with a wide range of applications, namely when dealing with affine nonlinear systems, which provide geometric control unification with asymptotic stability in some cases. A common scenario that comes from optimal control involves the minimization of integral cost functionals. Issues like asymptotic stability or even tracking to the desired reference signal have always been the main limitations. In this study, the main theorem allows the solution of optimal control problems with no-integral terms, in other words tracking problems with input/state constraints, providing closed-loop controllers. A DC motor with a pendulum in upright position is an example of an application for which singular optimal control is tested in this study. The results confirm both asymptotic stability and optimal tracking with an accuracy of 95%. The main contributions of this study include an optimal closed-loop controller with no mixed initial/final conditions, input/state constraints, asymptotic stability guarantee, a strong connection with geometric tools and finally the possibility to generalize to systems with multiple inputs. As a conclusion, general nonlinear control systems can be included in the optimal control methodology presented in this study including input/state constraints. Due to the lack of integral terms, the problem can be solved in closed form by using an optimal closed-loop controller.

Systematic application of Pontryagin's maximum principle to a general class of nonlinear mixed boundary value control problems yields necessary conditions for optimal control. Also, the features of certain pathological situations including the abnormal case and singular control are clearly revealed in this treatment.

In this paper a class of non-linear hereditary control systems is considered and necessary conditions for optimality of the initial data {u(t),φ(t)/t∊[∝,t0]} and the control {u(t)/t∝[t0,t1]} are derived. The results are direct extensions of Pontryagin's Maximum Principle. An example demonstrates the effect of the optimal selection of the initial data.

We study the problem of state transition on a finite time interval with minimal energy supply for linear port-Hamiltonian systems. While the cost functional of minimal energy supply is intrinsic to the port-Hamiltonian structure, the necessary conditions of optimality resulting from Pontryagin's maximum principle may yield singular arcs. The underlying reason is the linear dependence on the control, which makes the problem of determining the optimal control as a function of the state and the adjoint more complicated or even impossible. To resolve this issue, we fully characterize regularity of the (differential-algebraic) optimality system by using the interplay of the cost functional and the dynamics. In case of the optimality DAE being characterized by a regular matrix pencil, we fully determine the control on the singular arc. In case of singular matrix pencils of the optimality system, we propose an approach to compute rank-minimal quadratic perturbations of the objective such that the optimal control problem becomes regular. We illustrate the applicability of our results by a general second-order mechanical system and a discretized boundary-controlled heat equation.

SummaryLimited bandwidth and limited saturation in actuators are practical concerns in control systems. Mathematically, these limitations manifest as constraints imposed on the control actions, their rates of change, and, more generally, the global behavior of their paths. While the problem of actuator saturation has been studied extensively, little attention has been devoted to the problem of actuators having limited bandwidth. While attempts have been made in the direction of incorporating frequency constraints on state‐action trajectories before, rate constraints on the control at the design stage have not been studied extensively in the discrete‐time regime. This article contributes toward filling this lacuna: We establish a new discrete‐time Pontryagin maximum principle with rate constraints imposed on the control trajectories and derive first‐order necessary conditions for optimality. A brief discussion on the existence of optimal control is also included.

We study a model for the exploitation of renewable stocks developed in [5]. In this particular control problem, the control law contains a measurable and an impulsive control component. We formulate Pontryagin's maximum principle for this kind of control problems (see [3]), proving first order necessary conditions of optimality. Manipulating the correspondent Lagrange multipliers we manage to define two special switch functions, that allow the complete description the optimal trajectories and control policies, for all possible initial conditions in the phase plane.

Dengue remains a major public health threat in tropical and subtropical regions. We develop a general human–vector SEIR–SEI optimal‐control framework that integrates four time‐dependent interventions: public awareness/behavioral protection, , enhanced clinical management , adulticide spraying , and larval‐source reduction/larvicide . The controls act by reducing effective human–vector contact, increasing human recovery, increasing adult mosquito mortality, and suppressing vector recruitment, respectively. The objective is to minimize a weighted sum of infectious burden in humans and vectors and the quadratic costs of implementation over a finite horizon, subject to epidemiological dynamics and standard control bounds 0 ≤ u i ( t ) ≤ 1. Using Pontryagin's Maximum Principle, we derive the necessary conditions for optimality and solve the resulting two‐point boundary value problem numerically. Numerical simulations conducted in MATLAB, calibrated with real data from Bangladesh, perform uncertainty and sensitivity analyses around the basic reproduction number and key transmission, and control parameters reveal that Strategy‐4, which includes public awareness, treatment, and insecticide spraying (), is the most effective and cost‐efficient approach. This strategy reduces the time to disease elimination from over 100 days to approximately 74.7 days, achieving a 72.66% faster reduction than natural decay. The findings demonstrate that the proposed control strategies can significantly curb the progression of dengue and support targeted public health interventions to manage outbreaks effectively.

One of the major obstacles against succesful chemotherapy of cancer is the emergence of resistance of cancer cells to cytotoxic agents. Applying optimal control theory to mathematical models of cell cycle dynamics can be a very efficient method to understand and, eventually, overcome this problem. Results that have been hitherto obtained have already helped to explain some observed phenomena, concerning dynamical properties of cancer populations. Because of recent progress in understanding the way in which chemotherapy affects cancer cells, new insights and more precise mathematical formulation of control problem, in the meaning of finding optimal chemotherapy, became possible. This, together with a progress in mathematical tools, has renewed hopes for improving chemotherapy protocols. In this paper we consider a population of neoplastic cells stratified into subpopulations of cells of different types. Due to the mutational event a sensitive cell can acquire a copy of the gene that makes it resistant to the agent. Likewise, the division of resistant cells can result in the change of the number of gene copies. We convert the model in the form of an infinite dimensional system of ordinary differential state equations discussed in our previous publications (see e.g. Swierniak etal., 1996b; Polariski etal., 1997; Swierniak etaL, 1998c), into the integro‐differential form. It enables application of the necessary conditions of optimality given by the appropriate version of Pontryagin′s maximum principle, e.g. (Gabasov and Kirilowa, 1971). The performance index which should be minimized combines the negative cumulated cytotoxic effect of the drug and the terminal population of both sensitive and resistant neoplastic cells. The linear form of the cost function and the bilinear form of the state equation result in a bang‐bang optimal control law. To find the switching times we propose to use a special gradient algorithm developed similarly to the one applied in our previous papers to finite dimensional problems (Duda 1994; 1997).

This paper is the second of two in which the system studied has a plant which consists of two integrators with a saturable control input, and the performance criterion which is to be minimized is integral-error-squared. In the first paper the known optimum controller was shown to satisfy Pontryagin's maximum principle. In the present paper further verification of the optimum system is obtained by showing that the best performance function satisfies Bellman's partial differential equation. It is found that Bellman's equation is even satisfied for points on the switching curve, Results are also given for other performance criteria.

The spread of a virus or the outbreak of an epidemic are natural examples of stochastic processes. Understanding the epidemic dynamics, and finding out efficient techniques to control it, is a challenging issue. This paper investigates the optimal use of intervention strategies to mitigate the spread of infectious diseases. Classical mathematical descriptions of such phenomenon include various branching processes such as the SIR (Susceptible-Infected-Recovered). One reason for mathematical modelling is to analyze and predict the extent of emerging diseases and develop proposed control measures. The quadratic regulator format was used to formulate the optimal control problem, and two different optimal control techniques were investigated: Single Network Adaptive Critic (SNAC), which is a direct application of reinforcement learning theory to the optimality necessary conditions, and Approximate Sequence Riccati Equation (ASRE), which is a global optimal feedback control technique for general nonlinear systems with nonquadratic performance criteria. According to the results obtained during simulations, we claim that the proposed model and control strategy can be considered a good candidate to study viral spreading in the world. Also, the result shows that the Single Network Adaptive Critic is more accurate than the Approximate Sequence Riccati Equation.

In this paper, the authors consider a class of stochastic systems described by Ito differential equations for which both controls and parameters are to be chosen optimally with respect to a certain performance index over a fixed time interval. The controls to be optimized depend only on partially observed current states as in a work of Fleming. However, he considered, instead, a problem of optimal control of systems governed by stochastic Ito differential equations with Markov terminal time. The fixed time problems usually give rise to the Cauchy problems (unbounded domain) whereas the Markov time problems give rise to the first boundary value problems (bounded domain). This fact makes the former problems relatively more involved than the latter. For the latter problems, Fleming has reported a necessary condition for optimality and an existence theorem of optimal controls. In this paper, a necessary condition for optimality for both controls and parameters combined together is presented for the former problems.

Dynamic models which take the form of coupled set of differential and algebraic equations (DAEs) are widely used in process systems engineering. Necessary conditions of optimality for optimal control problems involving such models are derived. Interestingly, an optimality condition in the form of a strong maximum principle is shown to be valid only under a convexity hypothesis. However the authors prove a weak maximum principle even in the absence of convexity. >

Theoretical methods based on Pontryagin's maximum principle and Bellman's dynamic programming exist for determining optimal controls for nonlinear dynamical systems with state and control constraints. However, these methods only yield control inputs as functions of time (i.e., open loop controls). In practical applications, where robust feedback controllers are preferred, such solutions cannot be directly used. In this paper we propose methods for circumventing the above difficulties, and indicate how neural networks, trained off-line, may find application in the future in the design of online controllers for complex systems.

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula and Hamiltonian function on time scales, the necessary conditions of optimality are derived respectively. Some examples on continuous optimal control problems, discrete optimal control problems, mathematical programming and variational problems are also presented for demonstration.

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