An Optimal Control Problem with State Vector Measurement Errors

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

Optimal consumption and portfolio policies when asset prices follow a diffusion process

Reinforcement learning (RL) algorithms have been successfully applied to a range of challenging sequential decision making and control tasks. In this paper, we classify RL into direct and indirect RL according to how they seek the optimal policy of the Markov decision process problem. The former solves the optimal policy by directly maximizing an objective function using gradient descent methods, in which the objective function is usually the expectation of accumulative future rewards. The latter indirectly finds the optimal policy by solving the Bellman equation, which is the sufficient and necessary condition from Bellman's principle of optimality. We study policy gradient forms of direct and indirect RL and show that both of them can derive the actor-critic architecture and can be unified into a policy gradient with the approximate value function and the stationary state distribution, revealing the equivalence of direct and indirect RL. We employ a Gridworld task to verify the influence of different forms of policy gradient, suggesting their differences and relationships experimentally. Finally, we classify current mainstream RL algorithms using the direct and indirect taxonomy, together with other ones including value-based and policy-based, model-based and model-free.

SummaryThis paper considers optimal consensus control problem for unknown nonlinear multiagent systems (MASs) subjected to control constraints by utilizing event‐triggered adaptive dynamic programming (ETADP) technique. To deal with the control constraints, we introduce nonquadratic energy consumption functions into performance indices and formulate the Hamilton‐Jacobi‐Bellman (HJB) equations. Then, based on the Bellman's optimality principle, constrained optimal consensus control policies are designed from the HJB equations. In order to implement the ETADP algorithm, the critic networks and action networks are developed to approximate the value functions and consensus control policies respectively based on the measurable system data. Under the event‐triggered control framework, the weights of the critic networks and action networks are only updated at the triggering instants which are decided by the designed adaptive triggered conditions. The Lyapunov method is used to prove that the local neighbor consensus errors and the weight estimation errors of the critic networks and action networks are ultimately bounded. Finally, a numerical example is provided to show the effectiveness of the proposed ETADP method.

This paper deals with the problem of allocating attention among multiple tasks in a supervisory control system. The situation is modeled in the framework of a single server priority queueing system. Using Bellman's principle of optimality, a functional equation for the optimal, state dependent, nonpreemptive priority policy is obtained. The optimal policy is contrasted with an easily computable, suboptimal threshold policy, based on heavy traffic approximation. It is concluded that the utility of the optimal policy is significant at medium to high traffic intensities, and that the heavy traffic, suboptimal policy should be of considerable use in this regard. Several modifications of the single server model, as well as extensions to multiple, interacting sever models are pointed out.

This paper considers quickest detection scheme where the change in an underlying parameter influencing human decisions is to be detected by only observing the human decisions. Stemming from behavioral economics and mathematical psychology, we propose two generative models for the human decision maker. Namely, we consider an anticipatory decision making model and a quantum decision model. From a decision theoretic point of view, anticipatory models are time inconsistent, meaning that Bellman's principle of optimality does not hold. The appropriate formalism is thus the subgame Nash equilibrium. We show that the interaction between anticipatory agents and sequential quickest detection results in unusual (nonconvex) structure of the quickest change detection policy. In contrast the quantum decision model, despite its mathematical complexity, results in the typical convex quickest detection policy. The optimal quickest detection policy is shown to perform strictly worse than classical quickest detection for both models, via a Blackwell dominance argument. The model and structural results provided contribute to an understanding of the dynamics of human-sensor interfacing.

Development of the Schrödinger equation and Klein-Gordon equation via Bellman's principle of optimality

For Meyer type optimal control problems the principle of optimality states that every subarc of an optimal solution that extends to the final time furnishes again a solution to the same optimal control problem, only with different initial conditions. Through a simple counter example it is shown in this paper that this principle is not satisfied, in general, for solutions obtained from a standard collocation approach. Introduction The principle of optimality [1] is one of the most intuitive and yet most fundamental theorems in the theory of optimal control. Its basic statement is that for Meyer type optimal control problems every subarc of an optimal solution that extends to the final time furnishes again a solution to the same optimal control problem, only with different initial conditions. This theorem has significant implications. For example, it is the starting point for the derivation of the Hamilton-Jacobi-Bellman equations [2] and it is the underlying principle in Bellman's dynamic programming approach [1], [3]. The present paper shows that the principle of optimality is not satisfied, in general, for solutions obtained from a standard collocation approach[4]. For a simple example whose collocation solution can be generated analytically it is shown that further improvements are possible along an optimal collocation solution if the design parameters along a number of leading nodes are frozen and the remainder of the trajectory is re-optimized. I. Continuous Optimal Control Problem In this section of the paper we introduce a simple class of optimal control problems without state and control constraints. In Section 1.2 we present the associated necessary conditions for optimality [2], [5]. For ease of notation we assume that the right-hand side of the 'Supervising Engineer, Member AIAA, t Senior Engineer, Member AIAA, *17 Research Drive, Hampton, VA 23666. ^Graduate Student, Member AIAA, Department, 304 Town Engineering, Ames, IA 50011 equations of motion does not depend explicitly on time t. This does not represent a loss of generality, as it is always possible to transform explicit time dependence away through the introduction of an additional, timelike state variable. 1. Problem Formulation Let us consider the following class of optimal control problems: min «e(Pwc[*o,*/]), (1) subject to the conditions ±(t) = /(*(*),«(*)), tf.(z(*oMo) = 0, (2)

Minimizing Risk Models in Markov Decision Processes with Policies Depending on Target Values

In this paper, we consider a class of continuous-time, continuous-space stochastic optimal control problems. Using the Markov chain approximation method and recent advances in sampling-based algorithms for deterministic path planning, we propose a novel algorithm called the incremental Markov Decision Process to incrementally compute control policies that approximate arbitrarily well an optimal policy in terms of the expected cost. The main idea behind the algorithm is to generate a sequence of finite discretizations of the original problem through random sampling of the state space. At each iteration, the discretized problem is a Markov Decision Process that serves as an incrementally refined model of the original problem. We show that with probability one, (i) the sequence of the optimal value functions for each of the discretized problems converges uniformly to the optimal value function of the original stochastic optimal control problem, and (ii) the original optimal value function can be computed efficiently in an incremental manner using asynchronous value iterations. Thus, the proposed algorithm provides an anytime approach to the computation of optimal control policies of the continuous problem. The effectiveness of the proposed approach is demonstrated on motion planning and control problems in cluttered environments in the presence of process noise.

Motivated by the recent rapid growths of freemium services, we study freemium service providers’ dynamic operating policies to maximize their lifetime profits. We adopt a hybrid of the Bass diffusion model and the replicator equation from evolutionary game theory to capture the user base growth, and model the provider’s dynamic decisions as an optimal control problem. We analyze two variants of the freemium business model inspired by two prominent examples, Dropbox and Spotify, which, respectively, adopt the monetization strategies of limiting features and advertising (and its removal). We establish the optimal dynamic operating policies for the Dropbox-inspired ad-free freemium model and the Spotify-inspired ad-supported freemium model and identify their distinctive operational characteristics: the optimal policies as well as the sustainability of the Dropbox model critically depend on the network effect among users, whereas the optimal policies of the Spotify model show that advertising could be a more effective monetization strategy than limiting features. The latter observation is confirmed in a general model allowing both monetization strategies where the optimal policies consistently adopt advertising without limiting features. We then show that the freemium model significantly outperforms the ad-supported free model in weak advertising markets and that offering multiple tiers of premium subscription is unnecessary and confirm our core insights’ robustness under a generalized network effect, customized advertisements, and liquidity constraints. Our study establishes the versatility and value of the freemium model to suit distinct needs of a service in different stages from growth to monetization, and provides guidelines for service providers adopting this business model.

In this paper, we solve an optimal reinsurance problem in the mathematical finance area. We assume that the surplus process of the insurance company follows a controlled diffusion process and the constant interest rate is involved in the financial model. During the whole optimization period, the company has a choice to buy reinsurance contract and decide the reinsurance retention level. Meanwhile, the bankruptcy at the terminal time is not allowed. The aim of the optimization problem is to minimize the distance between the terminal wealth and a given goal by controlling the reinsurance proportion. Using the stochastic control theory, we derive the Hamilton-Jacobi-Bellman equation for the optimization problem. Via adopting the technique of changing variable as well as the dual transformation, an explicit solution of the value function and the optimal policy are shown. Finally, several numerical examples are shown, from which we find several main factors that affect the optimal reinsurance policy.

Optimal radio access technology selection on heterogeneous networks

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.

Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates