Lipschitzian Regularity of the Minimizing Trajectories for Nonlinear Optimal Control Problems

We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions can we assure optimal controls are bounded? This question is related to one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial in closing the gap between the conditions arising in existence theory and necessary optimality conditions. Rewriting the Lagrange problem in a parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle to the latter problem and the Lipschitzian regularity conditions for the original problem. Under the standard hypotheses of coercivity of the existence theory, the conditions imply that the optimal controls are essentially bounded, assuring the applicability of the classical necessary optimality conditions like the Pontryagin maximum principle. The result extends previous Lipschitzian regularity results to cover optimal control problems with general nonlinear dynamics.     

Point-to-point control near heteroclinic orbits: Plant and controller optimality conditions

In this paper we consider the simultaneous optimization of the controller and plant in a one degree-of-freedom system. In particular we are interested in optimal trajectories between fixed points connected by heteroclinic orbits. We find that designing the plant dynamics to have a heteroclinic connection between target states enables a low energy transfer between the states. We use a nested optimization strategy to find the optimal plant dynamics and control effort for the transition. Additionally, we uncover plant optimality conditions which reduce the complexity of the optimization.     

Optimisation of strain selection in evolutionary continuous culture

In this work, we study a minimal time control problem for a perfectly mixed continuous culture with n ≥ 2 species and one limiting resource. The model that we consider includes a mutation factor for the microorganisms. Our aim is to provide optimal feedback control laws to optimise the selection of the species of interest. Thanks to Pontryagin's Principle, we derive optimality conditions on optimal controls and introduce a sub-optimal control law based on a most rapid approach to a singular arc that depends on the initial condition. Using adaptive dynamics theory, we also study a simplified version of this model which allows to introduce a near optimal strategy.     

A frequency‐constrained geometric Pontryagin maximum principle on matrix Lie groups

We present a geometric discrete‐time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the control trajectories in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first‐order necessary conditions for optimality and leads to two‐point boundary value problems that may be solved by numerical techniques to arrive at optimal trajectories. We demonstrate our theoretical results with numerical simulations on the optimal trajectory generation of a wheeled inverted pendulum and an attitude control problem of a spacecraft on the Lie group SO(3).     

Optimal controls of multidimensional modified Swift–Hohenberg equation

In this paper, we deal with the optimal control problem governed by multidimensional modified Swift–Hohenberg equation. After showing the relationship between the control problem and its approximation, we derive the optimality conditions for an optimal control of our original problem by using one of the approximate problems.     

A pseudospectral method for the optimal control of constrained feedback linearizable systems

We consider the optimal control of feedback linearizable dynamical systems subject to mixed state and control constraints. In general, a linearizing feedback control does not minimize the cost function. Such problems arise frequently in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls. In this paper, we consider a pseudospectral (PS) method to compute optimal controls. We prove that a sequence of solutions to the PS-discretized constrained problem converges to the optimal solution of the continuous-time optimal control problem under mild and numerically verifiable conditions. The spectral coefficients of the state trajectories provide a practical method to verify the convergence of the computed solution. The proposed ideas are illustrated by several numerical examples.     

Analogue of the Kelley condition for optimal systems with retarded control

In this paper, we consider an optimal control problem with retarded control and study a larger class of singular (in the classical sense) controls. The Kelley and equality type optimality conditions are obtained. To prove our main results, we use the Legendre polynomials as variations of control.     

An optimal regulation strategy with disturbance rejection for energy management of hybrid electric vehicles

The energy management problem of finding the optimal split between the different sources of energy in a charge-sustaining parallel HEV, ensuring stability and optimality with respect to a performance objective (fuel consumption minimization over a driving cycle), is addressed in this paper. The paper develops a generic stability and optimality framework within which the energy management problem is cast in the form of a nonlinear optimal regulation (with disturbance rejection) problem and a control Lyapunov function is used to design the control law. Two theorems ensuring optimality and asymptotic stability of the energy management strategy are proposed and proved. The sufficient conditions for optimality and stability are used to derive an analytical expression for the control law as a function of the battery state of charge/state of energy and system parameters. The control law is implemented in a simplified backward vehicle simulator and its performance is evaluated against the global optimal solution obtained from dynamic programming. The strategy performs within 4% of the benchmark solution while guaranteeing optimality and stability for any driving cycle.     

Optimal control of switching systems

This paper considers an optimal control problem for a switching system. For solving this problem we do not make any assumptions about the number of switches nor about the mode sequence, they are determined by the solution of the problem. The switching system is embedded into a larger family of systems and the optimization problem is formulated for the latter. It is shown that the set of trajectories of the switching system is dense in the set of trajectories of the embedded system. The relationship between the two sets of trajectories (1) motivates the shift of focus from the original problem to the more general one and (2) underlies the engineering relevance of the study of the second problem. Sufficient and necessary conditions for optimality are formulated for the second optimization problem. If they exist, bang-bang-type solutions of the embedded optimal control problem are solutions of the original problem. Otherwise, suboptimal solutions are obtained via the Chattering Lemma.     

A Youla–Kučera parameterization approach to output feedback relatively optimal control

This paper presents a continuous time solution to the problem of designing a relatively optimal control, precisely, a dynamic control which is optimal with respect to a given initial condition and is stabilizing for any other initial state. This technique provides a drastic reduction of the complexity of the controller and successfully applies to systems in which (constrained) optimality is necessary for some “nominal operation” only. The technique is combined with a pole assignment procedure. It is shown that once the closed-loop poles have been fixed and an optimal trajectory originating from the nominal initial state compatible with these poles is computed, a stabilizing compensator which drives the system along this trajectory can be derived in closed form. There is no restriction on the optimality criterion and the constraints. The optimization is carried out over a finite-dimensional parameterization of the trajectories. The technique has been presented for state feedback. We propose here a technique based on the Youla–Kučera parameterization which works for output feedback. The main result is that we provide conditions for solvability in terms of a set of linear algebraic equations.     

A local feedback synthesis of time-optimal stabilizing controls in dimension three

Under generic conditions a local feedback synthesis for the problem of time-optimally stabilizing an equilibrium point in dimension three is constructed. There exist two surfaces which are glued together along a singular are on which the optimal control is singular. Away from these surfaces the optimal controls are piecewise constant with at most two switchings. Bang-bang trajectories with two switchings but different switching orders intersect in a nontrivial cut-locus and optimality of trajectories ceases at this cut-locus. The construction is based on an earlier result by Krener and Schättler which gives the precise structure of the small-time reachable set for an associated system to which time has been added as an extra coordinate.     

Optimization of the separation of two species in a chemostat

In this work, we study a two species chemostat model with one limiting substrate, and our aim is to optimize the selection of the species of interest. More precisely, the objective is to find an optimal feeding strategy in order to reach in minimal time a target where the concentration of the first species is significantly larger than the concentration of the other one. Thanks to the Pontryagin Maximum Principle, we introduce a singular feeding strategy which allows to reach the target, and we prove that the feedback control provided by this strategy is optimal whenever initial conditions are chosen in the invariant attractive manifold of the system. The optimal synthesis of the problem in presence of more than one singular arc is also investigated.     

Simultaneous dynamic optimization: A trajectory planning method for nonholonomic car-like robots

Trajectory planning in robotics refers to the process of finding a motion law that enables a robot to reach its terminal configuration, with some predefined requirements considered at the same time. This study focuses on planning the time-optimal trajectories for car-like robots. We formulate a dynamic optimization problem, where the kinematic principles are accurately described through differential equations and the constraints are strictly expressed using algebraic inequalities. The formulated dynamic optimization problem is then solved by an interior-point-method-based simultaneous approach. Compared with the prevailing methods in the field of trajectory planning, our proposed method can handle various user-specified requirements and different optimization objectives in a unified manner. Simulation results indicate that our proposal efficiently deals with different kinds of physical constraints, terminal conditions and collision-avoidance requirements that are imposed on the trajectory planning mission. Moreover, we utilize a Hamiltonian-based optimality index to evaluate how close an obtained solution is to being optimal.     

Sufficient optimality conditions for Pontryagin extremals

L₁-local optimality of a given control (·) in an optimal control problem for an affine control system with bounded controls is investigated. Starting from the Pontryagin Maximum Principle, which is a first-order necessary optimality condition, we develop it in two directions: (1) extending the notions of 1st and 2nd variations of the system along (·), we obtain 1st and 2nd-order sufficient optimality conditions for bang-bang Pontryagin extremals; (2) developing Legendre-Jacobi-Morse-type results for the extended second variation we obtain 2nd-order sufficient optimality conditions for general (bang-bang-singular) type of Pontryagin extremals.     

Nonstandard duality and nonlocal necessary optimality conditions in nonconvex optimal control problems

To partially implement the idea of considering nonlinear optimal control problems immediately on the set of Pontryagin extremals (or on quasiextremals if the optimal solution does not exist), we introduce auxiliary functions of canonical variables, which we call bipositional, and the corresponding modified Lagrangian for the problem. The Lagrangian is subject to minimization on the trajectories of the canonical system from the Maximum Principle. This general approach is further specialized for nonconvex problems that are linear in state, leading to a nonstandard dual optimal control problem on the trajectories of the adjoint system. Applying the feedback minimum principle to both original and dual problems, we have obtained a pair of necessary optimality conditions that significantly strengthen the Maximum Principle and admit a constructive realization in the form of an iterative problem solving procedure. The general approach, optimality features, and the iterative solution procedure are illustrated by a series of examples.     

Fractional optimal control problems with both integer-order and Atangana–Baleanu Caputo derivatives

This article considers fractional optimal control problems (FOCPs) including both integer-order and Atangana–Baleanu Caputo derivatives. First, the existence and uniqueness of the solution of a fractional Cauchy problem is given. Then, applying calculus of variations and Lagrange multiplier method, we present necessary optimality conditions of FOCPs and sufficient optimality conditions are also given under some assumptions. Next, a collection method is developed to derive numerical solutions by using shifted Legendre polynomials. Finally, error estimate of numerical solutions is also provided, and numerical examples further show the accuracy and feasibility of our method.     

Nonlinear overtaking optimal control: sufficiency, stability, andapproximation

The authors consider infinite-horizon nonlinear optimal control problems with unbounded performance indexes using notions of overtaking optimality. In the framework of optimal control, new sufficient conditions of Caratheodory-Hamilton-Jacobi type for both weak overtaking and overtaking optimality are presented and compared with known results from the calculus of variations setting. Using local structural properties of a related control Hamiltonian, conclusions are drawn on aspects of existence of overtaking and weak overtaking optimal controls, stability of optimal trajectories, and approximation of weak overtaking optimal controls. Both similarities and differences among alternative approaches to overtaking, weak overtaking, and strong optimal control problems are discussed. The formulation and results are illustrated in the classical linear-quadratic tracking problem and in an optimization approach to gain scheduling     

Time-domain decomposition of optimal control problems for the wave equation

We consider the problem of boundary optimal control of a wave equation with boundary dissipation by the way of time-domain decomposition of the corresponding optimality system. We develop an iterative algorithm which shows that the decomposed optimality system corresponds to local-in-time optimal control problems which can be treated in parallel. We show convergence of the algorithm. Finally, we provide a time discretization which is reminiscent of an instantaneous control scheme. We thereby also contribute to the problem of convergence of such schemes.     

A risk‐sensitive stochastic maximum principle for fully coupled forward‐backward stochastic differential equations with applications

In this paper, we are interested in the problem of optimal control where the system is given by a fully coupled forward‐backward stochastic differential equation with a risk‐sensitive performance functional. As a preliminary step, we use the risk neutral which is an extension of the initial control system where the admissible controls are convex, and an optimal solution exists.Then, we study the necessary as well as sufficient optimality conditions for risk sensitive performance. At the end of this work, we illustrate our main result by giving an example that deals with an optimal portfolio choice problem in financial market, specifically the model of control cash flow of a firm or project where, for instance, we can set the model of pricing and managing an insurance contract.     

Optimal Population Transfers in a Quantum System for Large Transfer Time

Transferring a quantum system to a final state with given populations is an important problem with applications to quantum chemistry and atomic physics. In this paper, we consider such transfers that minimize L² the norm of the control. This problem is challenging, both analytically and numerically. With the exception of the simplest cases, there is no general understanding of the nature of optimal controls and trajectories. We find that, by examining the limit of large transfer times, we can uncover such general properties. In particular, for transfer times large with respect to the time scale of the free dynamics of the quantum system, the optimal control is a sum of terms, each being a Bohr frequency sinusoid modulated by a slow amplitude, i.e., a profile that changes considerably only on the scale of the transfer time. Moreover, we show that the optimal trajectory follows a ldquomeanrdquo evolution modulated by the fast free dynamics of the system. The calculation of the ldquomeanrdquo optimal trajectory and the slow control profiles is done via an ldquoaveragedrdquo two-point boundary value problem that we derive and which is much easier to solve than the one expressing the necessary conditions for optimality of the original optimal transfer problem.