Necessary Conditions For Optimality Research Articles

Robust Optimality and Duality for Nonsmooth Multiobjective Programming Problems with Vanishing Constraints Under Data Uncertainty

This article investigates robust optimality conditions and duality results for a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty (UNMPVC). Mathematical programming problems with vanishing constraints constitute a distinctive class of constrained optimization problems because of the presence of complementarity constraints. Moreover, uncertainties are inherent in various real-life problems. The aim of this article is to identify an optimal solution to an uncertain optimization problem with vanishing constraints that remains feasible in every possible future scenario. Stationary conditions are necessary conditions for optimality in mathematical programming problems with vanishing constraints. These conditions can be derived under various constraint qualifications. Employing the properties of convexificators, we introduce generalized standard Abadie constraint qualification (GS-ACQ) for the considered problem, UNMPVC. We introduce a generalized robust version of nonsmooth stationary conditions, namely a weakly stationary point, a Mordukhovich stationary point, and a strong stationary point (RS-stationary) for UNMPVC. By employing GS-ACQ, we establish the necessary conditions for a local weak Pareto solution of UNMPVC. Moreover, under generalized convexity assumptions, we derive sufficient optimality criteria for UNMPVC. Furthermore, we formulate the Wolfe-type and Mond–Weir-type robust dual models corresponding to the primal problem, UNMPVC.

NMPC‐Based Control of Underactuated Spacecraft: A Neighboring Extremal Approach Extended to SE(3)

ABSTRACTThe present work revolves around employing the benefits of neighboring extremal approach in the nonlinear model predictive control of mechanical systems evolving on SE(3). In the NMPC process, necessary conditions of optimality are extracted based on some discrete‐time version of the equations of motion referred as LGVI and the obtained TPBVP equations are solved using simplified sensitivity derivatives by means of an indirect shooting method. Taking into consideration that the repetitive optimization process of the NMPC is time consuming, making its implementation challenging, a method of neighboring extremal is proposed here for recalculating the responses of the systems in scenarios that face some unexpected changes in their parameters or are encountering some last‐minute re‐planning schemes. Based on the existing responses of the system to the first set of initial conditions, the reaction of the system to the altered set of initial conditions can be reconstructed without the need to solve the whole optimization process from scratch by utilizing the features of the NE method. A spacecraft model evolving on SE(3) with actuation constraints confirms the efficiency of the whole process in term of accuracy versus computation burden. Forasmuch as actuator failure is a probable yet important event in aerial/spatial missions, the performance of the control system in dealing with such situations is examined. The algorithm robustness and its capability to compensate the effects of the actuator loss is checked.

Energy optimal path planning for wheeled mobile robots with circular obstacle

AbstractThe energy optimal motion planning algorithm for a differential drive robot with a circular obstacle in the maneuver space is introduced in this article. Rather than the traditional unicycle model, a non‐canonical state space model of the kinematics of a wheeled mobile robot is used to formulate the optimal control problem. The necessary conditions for optimality are formally derived using calculus of variations, resulting in constraints for the costates when the inequality constraint become active, revealing that the controls are required to be continuous. The optimality conditions permit decomposing the motion planning problem in two independent segments: prior to and after activation of the inequality constraints. Analytical expressions for the evolving states and control are found to be a function of Jacobi elliptic functions, permitting reducing the optimal control problem to a root‐finding problem. A comprehensive parametric study is included to demonstrate the impact on the optimal trajectories by varying the radius of the obstacle. The study shows that there are three distinct maneuver strategies with respect to the size of the obstacle. Furthermore, by including entering and exit time as optimization variables, a methodology for generalizing the development to any given motion scenario is developed and numerically illustrated.

Identification of sparse nonlinear controlled variables for near‐optimal operation of chemical processes

AbstractFor optimal operation of chemical processes, the selection of controlled variables plays an important role. A previous proposal is to approximate the necessary conditions of optimality (NCO) as the controlled variables, such that process optimality is automatically maintained by tracking constant zero setpoints. In this paper, we extend the NCO approximation method by identifying sparse nonlinear controlled variables, motivated by the fact that simplicity is always favoured for practical implementations. To this end, the ‐regularization is employed to approximate the NCO, such that the controlled variables are maintained simple, even they are specified as nonlinear functions. The sparse controlled variables are solved using the proximal gradient method, implemented within a tailored Adam algorithm. Two case studies are provided to illustrate the proposed approach.

The maximum principle for lumped-distributed control systems.

This paper concerns the optimal control of lumped-distributed systems, an examplar of which is a mechanical system with rotating components connected by flexible rods. The underlying mathematical model is a controlled semilinear evolution equation, in which nonlinear terms involve a projection of the full state onto a finite dimensional subspace. We derive necessary conditions of optimality in the form of a maximum principle, for a problem formulation which involves pathwise and end-point constraints on the lumped components of the state variable. Perturbational methods are employed in the derivation, which allow for non-smooth data. A key step in the analysis is the reduction of the optimization problem with an infinite dimensional state space, to one with finite dimensional aspects. The computational implications of this reduction will be explored in future work. Necessary conditions for problems with state constraints and end-point constraints can only be achieved under rather strict compatibility hypotheses on the way the dynamics interact with the state constraint and end-point constraint sets, due the infinite dimensionality of the state space. This paper identifies a class of infinite dimensional problems, of engineering significance, for which necessary conditions of optimality can be derived, in the absence of any additional compatibility hypotheses.

Optimal control problems for a parabolic system inspired by a cancer therapy

In this paper, we consider optimal control problems for a parabolic system modeling a therapy, based on oncolytic viruses, for the glioma brain cancer. Using several techniques typical of functional analysis, we prove the global in time well posedness of the control model, the existence of optimal controls for specific objective functionals, which are natural for cancer therapies, and we derive necessary conditions for optimality.

Optimal Control Problem with Regular Mixed Constraints via Penalty Functions

AbstractBelow we derive necessary conditions of optimality for problems with mixed constraints (see Dmitruk in Control Cybern 38(4A):923–957, 2009) using the method of penalty functions similar to the one we previously used to solve optimization problems for control sweeping processes (see, e.g., De Pinho et al. in Optimization 71(11):3363–3381, 2022) and, more recently, to solve optimal control problems with pure state constraints (see De Pinho et al. in Syst Control Lett 188:105816, 2024). We intentionally consider a smooth case and the simplest boundary conditions; we consider global minimum and assume that the set of trajectories of the control system is compact. Based on our penalty functions approach we develop a numerical method admitting estimates for its parameters needed to achieve a given precision.

Discrete‐time Pontryagin maximum principle under rate constraints: Necessary conditions for optimality

Discrete‐time Pontryagin maximum principle under rate constraints: Necessary conditions for optimality

Stochastic maximum principle for partially observed optimal control problem of McKean–Vlasov FBSDEs with Teugels martingales

Abstract In this paper, we study the stochastic maximum principle for a partially observed optimal control problem of forward-backward stochastic differential equations (FBSDEs for short) of McKean–Vlasov type driven by both a family of Teugels martingales and an independent Brownian motion. The coefficients of the system and the cost functional depending on the state of the solution process as well as its probability law and the control variable. We establish partially observed necessary conditions of optimality for this system under assumption that the control domain is supposed to be convex. Our main result is based on Girsavov’s theorem and the derivatives with respect to probability law. As an application of the general theory, a partially observed linear-quadratic control problem of McKean–Vlasov type is studied in terms of stochastic filtering.

Non-degenerate necessary conditions for non-convex differential inclusions

This paper concerns dynamic optimization problems with pathwise state constraints, in which the dynamics are formulated as a differential inclusion. First order necessary conditions of optimality are available for these problems. Foremost among these conditions is the Euler Lagrange inclusion (combined with a transversality and Weierstrass conditions), which improves on earlier conditions such as Clarke's Hamiltonian inclusion. The most recent version of the Euler Lagrange inclusion (which we call the standard Euler Lagrange condition) has been validated under unrestrictive conditions, which allow the velocity sets of the differential inclusion to be non-convex. If the initial state is fixed and located in the boundary of the state constraint set then the Euler Lagrange inclusion conveys no information about minimizers. This is the degeneracy phenomenon for state constrained dynamic optimization. The non-degenerate necessary conditions literature provides additional conditions, to supplement the standard necessary conditions. These typically identify some properties of the boundary values of the maximized Hamiltonian which ensure non-triviality of the necessary conditions, under a simple controllability hypothesis. Non-degenerate conditions, incorporating the Euler Lagrange inclusion are currently available only under the hypothesis that the velocity sets are convex. Using new perturbation techniques, we derive a non-degenerate Euler Lagrange condition that is valid for non-convex velocity sets. These conditions bring into line non-degenerate optimality conditions for differential inclusion problems with those for problems involving controlled differential equations which have previously been validated for non-convex velocity sets.

Stochastic intervention control of mean-field jump system with noisy observation via L-derivatives with application to finance

In this paper, we investigate stochastic optimal intervention control of mean-field nonlinear random Poisson-jump-system with related noisy process. We derive the necessary conditions of optimality for partially observed optimal intervention control problems of mean-field type. The coefficients depend on the state of the solution process as well as of its probability distribution and the control variable. The proof of our main result is obtained by applying L-derivatives in the sense of Lions. In our control model, there are two models of jumps for the state process, the inaccessible ones which come from the random Poission process and the predictable ones which come from the intervention control. Finally, we apply our result to study conditional mean-variance portfolio selection problem with interventions, where the foreign exchange interventions are intended to contain excessive fluctuations in foreign exchange rates and to stabilize them.

Optimal control problem with state constraints via penalty functions

In this paper, we derive necessary conditions of optimality for problems with state constraints using the method of penalty functions similar to the one we previously used to solve optimization problems for control sweeping processes (see, e.g., de Pinho et al., 2022). Our aim is to provide a proof of the Maximum Principle that can be easily followed by those with basic knowledge of classical optimal control and elementary functional analysis. We intentionally consider a smooth case and the simplest boundary conditions; we consider global minimum and assume that the set of trajectories of the control system is compact.

Multiple-arc optimization of low-thrust Earth–Moon orbit transfers leveraging implicit costate transformation

This work focuses on minimum-time low-thrust orbit transfers from a prescribed low Earth orbit to a specified low lunar orbit. The well-established indirect formulation of minimum-time orbit transfers is extended to a multibody dynamical framework, with initial and final orbits around two distinct primaries. To do this, different representations, useful for describing orbit dynamics, are introduced, i.e., modified equinoctial elements (MEE) and Cartesian coordinates (CC). Use of two sets of MEE, relative to either Earth or Moon, allows simple writing of the boundary conditions about the two celestial bodies, but requires the formulation of a multiple-arc trajectory optimization problem, including two legs: (a) geocentric leg and (b) selenocentric leg. In the numerical solution process, the transition between the two MEE representations uses CC, which play the role of convenient intermediate, matching variables. The multiple-arc formulation at hand leads to identifying a set of intermediate necessary conditions for optimality, at the transition between the two legs. This research proves that a closed-form solution to these intermediate conditions exists, leveraging implicit costate transformation. As a result, the parameter set for an indirect algorithm retains the reduced size of the typical set associated with a single-arc optimization problem. The indirect heuristic technique, based on the joint use of the necessary conditions and a heuristic algorithm (i.e., differential evolution in this study) is proposed as the numerical solution method, together with the definition of a layered fitness function, aimed at facilitating convergence. The minimum-time trajectory of interest is sought in a high-fidelity dynamical framework, with the use of planetary ephemeris and the inclusion of the simultaneous gravitational action of Sun, Earth, and Moon, along the entire transfer path. The numerical results unequivocally prove that the approach developed in this research is effective for determining minimum-time low-thrust Earth–Moon orbit transfers.

A Caputo fractional derivative dynamic model of hepatitis E with optimal control based on particle swarm optimization

Hepatitis E, as a zoonotic disease, has been a great challenge to global public health. Therefore, it has important research value and practical significance for the transmission and control of hepatitis E virus (HEV). In the exploration of infectious disease transmission dynamics and optimal control, mathematical models are often applied. Among them, the fractional differential model has become an important and practical tool because of its good memory and genetic characteristics. In this paper, an HEV propagation dynamic model is constructed by the Caputo fractional derivative. First, the properties of the model are analyzed, including the existence, non-negativity, boundedness, and stability of the equilibrium points. Then, from the perspective of fractional optimal control (FOC), control measures were proposed, including improving the awareness and prevention of hepatitis E among susceptible people, strengthening the treatment of infected people, and improving environmental hygiene. Then, an FOC model of HEV was constructed. After analyzing the necessary conditions for optimality, the particle swarm optimization is introduced to optimize the control function. In addition, four control strategies are applied. Finally, the numerical simulation is completed by the fractional Adams–Bashforth–Moulton prediction correction algorithm. The four strategies and no control were compared and analyzed. The numerical simulation results of different fractional orders are also compared and analyzed. The results illustrate that the optimal strategy, compared with no control, reduces the HEV control time by nearly 60 days. Therefore, this method would contribute to the study of HEV transmission dynamics and control mechanisms, thus contributing to the development of global public health.

Controlled traveling profiles for models of invasive biological species

We consider a family of controlled reaction-diffusion equations, describing the spatial spreading of an invasive biological species. For a given propagation speed c, we seek a control with minimum cost, which achieves a traveling profile with speed c. Since our goal is to slow down or even reverse the contamination, we always assume c > c∗, where c∗ is the speed of an uncontrolled traveling profile. For various nonlinear models, the existence of an optimal control is proved, together with necessary conditions for optimality. In the last section we study a case where the wave speed cannot be modified by any control with finite cost. The present analysis is motivated by the recent results in [8, 9], showing how a control problem for a reaction-diffusion equation can be approximated by a simpler problem of optimal control of a moving set.

New Adaptive Mesh Refinement Strategy for Entry Guidance via Sequential Convex Programming

This study focuses on the entry guidance problem, which is a highly nonlinear and constrained optimal control problem. The entry guidance problem is formulated as a nonlinear trajectory optimization problem with six-degrees-of-freedom dynamics, path constraints, and boundary constraints. The formulated problem is discretized and solved using sequential convex programming (SCP) with successive linearization. However, during the discretization process, conventional collocation methods encounter a challenge with high-frequency jittering in the control and bank angle profiles near active path constraints. Because the considered problem especially involves tight path constraints, the solution jittering issue becomes severe. Therefore, this study proposes a novel mesh refinement strategy to address this problem. Unlike conventional mesh refinement strategies based on interpolation error, the proposed approach resolves the convergence and solution jittering issue of the SCP framework by linking the direct and indirect methods with the costate estimation under successive linearization. Along with the development, we introduce a novel costate estimation method and provide proof of its validity. We present numerical simulation results of the proposed method and conduct a comparison with conventional approaches. The results obtained demonstrate successful mitigation of the solution jittering issue and a significant improvement in solution accuracy regarding compliance with the necessary conditions of optimality.

Real-time optimal control for attitude-constrained solar sailcrafts via neural networks

This work is devoted to generating optimal guidance commands in real time for attitude-constrained solar sailcrafts in coplanar circular-to-circular interplanetary transfers. Firstly, a nonlinear optimal control problem is established, and the necessary conditions for optimality are derived by Pontryagin’s Minimum Principle. Under some mild assumptions, the attitude constraints are rewritten as control constraints, which are replaced by a saturation function so that a parameterized system is formulated. This allows one to generate an optimal trajectory via solving an initial value problem, making it efficient to collect a dataset containing optimal samples, which are essential for training Neural Networks (NNs) to achieve real-time implementation. However, the optimal guidance command may suddenly change from one extreme to another, resulting in discontinuous jumps that generally impair the NN’s approximation performance. To address this issue, we use two co-states that the optimal guidance command depends on, to detect discontinuous jumps. A procedure for preprocessing these jumps is then established, thereby ensuring that the preprocessed guidance command remains smooth everywhere. Meanwhile, the sign of one co-state is found to be sufficient to revert the preprocessed guidance command back into the original optimal guidance command. Furthermore, three NNs are built and trained offline, and they cooperate to precisely generate the optimal guidance command in real time. Finally, numerical simulations are presented to demonstrate the developments of the paper.

In search of maximum non-overlapping codes

AbstractNon-overlapping codes are block codes that have arisen in diverse contexts of computer science and biology. Applications typically require finding non-overlapping codes with large cardinalities, but the maximum size of non-overlapping codes has been determined only for cases where the codeword length divides the size of the alphabet, and for codes with codewords of length two or three. For all other alphabet sizes and codeword lengths no computationally feasible way to identify non-overlapping codes that attain the maximum size has been found to date. Herein we characterize maximal non-overlapping codes. We formulate the maximum non-overlapping code problem as an integer optimization problem and determine necessary conditions for optimality of a non-overlapping code. Moreover, we solve several instances of the optimization problem to show that the hitherto known constructions do not generate the optimal codes for many alphabet sizes and codeword lengths. We also evaluate the number of distinct maximum non-overlapping codes.

Аналог принципа максимума Понтрягина в одной задаче оптимального управления с переменной структурой

In this paper we consider one optimal control problem with distributed parameters described in two different domains by two Goursat-Darboux systems under the assumption that the control domains are arbitrary. The quality criterion is a terminal type functional. Based on a modified version of the increment method, a necessary condition for optimality is proved in the form of an analogue of Pontryagin's maximum principle.

Optimal tracking control of an Autonomous Underwater Vehicle: A PMP approach

This paper addresses the optimal tracking control problem of an Autonomous Underwater Vehicle (AUV) using Pontryagin’s Minimum Principle (PMP). The formulation uses a choice of Hamiltonian function using the desired control objective with AUV dynamics acting as dynamic constraints. We first develop PMP based on AUV kinematics and then extend it to the dynamics to arrive at optimal thrusts and moments. Necessary conditions for optimality are derived for both the models using PMP, which results in optimal trajectories that simultaneously minimize the tracking error as well as the control cost, thereby arriving at energy optimality. It was observed that the adjoint variables (costates) are indeed the momenta in the inertial and body-fixed frames. At the kinematic level, this forms a stable solution. The developed methodology is applied to both 2D and 3D AUV model with disturbances due to inputs and ocean currents. Numerical simulations are carried out with the derived control laws for a given trajectory tracking target. Quantitative evaluation of the performance and comparison of the controller is done using Mean Square Error(MSE) and Total Variation(TV) measures. The proposed control laws are found to achieve the desired control objectives.