Necessary Conditions For Optimality Research Articles

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.

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.

Nonregular multiobjective problems: optimality conditions

Necessary and sufficient conditions for multiobjective problems with both equality constraints and inequality constraints are presented. These conditions are applied when the constraints of the problem do not necessarily satisfy any regularity assumptions and they are based on the concept of p-regularity, introduced by Tret'yakov (1983, Necessary conditions for optimality of pth order. Control Optim.;28–35). We introduce the notion of generalized Mangasarian-Fromovitz constraint qualification. This new concept is used to obtain our results.

Minimum-fuel two-impulse transfer between coplanar noncoaxial elliptical orbits

This paper investigates minimum-fuel two-impulse transfer between general coplanar noncoaxial elliptical orbits in a Newtonian gravitational field. First, without any restriction on the transfer time and boundary conditions, a necessary condition of optimality for the minimum-fuel solutions is derived via the primer vector analysis and thus, a simplified optimization model is presented. Then, a homotopic approach based algorithm is proposed to obtain the minimum-fuel solutions, which can start automatically from the analytical generalized Hohmann transfers. Finally, accuracy and efficiency of the algorithm are illustrated via different examples. Symmetric property of the minimum-fuel solutions is drawn. Special cases where the two-impulse solutions degenerate to one-impulse solutions are analyzed. Further discussions are given to explore the global optimality of the obtained minimum-fuel solutions.

A shape parameter insensitive CRBFs‐collocation for solving nonlinear optimal control problems

AbstractIn this work, we introduce a Coupled Radial Basis Functions collocation (CRBFs‐Coll) approach for solving optimal control problems. CRBFs are real‐valued Radial Basis Functions (RBFs) augmented with a conical spline. They show insensitivity to the shape parameter resulting in robust behavior in function approximation. The method is applied to classical nonlinear optimal control problems: Zermelo's problem, a Duffing oscillator with various boundary conditions, and a nonlinear pendulum on a cart problem. The nonlinear optimal control problem is solved by deriving the necessary conditions for optimality followed by collocation using CRBFs as basis functions for approximating the two‐point boundary value problem (TPBVP). The resulting system of nonlinear algebraic equations (NAEs) is then solved using a standard solver. Unlike existing methods that rely on nodal distributions that are denser at the boundaries to obtain a solution, the present CRBFs‐Coll approach is capable of solving the optimal control problem on uniform nodes without the need for interpolation. The present CRBFs‐Coll approach is shown to be simple to implement and provides accurate results over an equidistant nodal distribution while maintaining a continuous representation of the control and states.

Dynamic analysis and control of a rice-pest system under transcritical bifurcations.

A decision model is developed by adopting two control techniques, combining cultural methods and pesticides in a hybrid approach. To control the adverse effects in the long term and to be able to evaluate the extensive use of pesticides on the environment and nearby ecosystems, the novel decision model assumes the use of pesticides only in an emergency situation. We, therefore, formulate a rice-pest-control model by rigorously modelling a rice-pest system and including the decision model and control techniques. The model is then extended to become an optimal control system with an objective function that minimizes the annual losses of rice by controlling insect pest infestations and simultaneously reduce the adverse impacts of pesticides on the environment and nearby ecosystems. This rice-pest-control model is verified by analysis, obtains the necessary conditions for optimality, and confirms our main results numerically. The rice-pest system is verified by stability analysis at equilibrium points and shows transcritical bifurcations indicative of acceptable thresholds for insect pests to demonstrate the pest control strategy.

Approximate Solutions for Optimal Control of Fixed Boundary Value Problems Using Variational and Minimum Approaches

The optimal control is the process of finding a control strategy that extreme some performance index for a dynamic system (partial differential equation) over the class of admissibility. The present work deals with a problem of fixed boundary with a control manipulated in the structure of the partial differential equation. An attractive computational method for determining the optimal control of unconstrained linear dynamic system with a quadratic performance index is presented. In the proposed method the difference between every state variable and its initial condition is represented by a finite - term polynomial series, this representation leads to a system of linear algebraic equations which represents the necessary condition of optimality. The linear algebraic system is solved by using two approaches namely the variational iteration method and the minimization approach for unconstrained optimization problem with estimation of gradient and Hessian matrix. These approaches are illustrated by two application examples.

Application of the Hamiltonian System in Deriving Solution to Dynamic System of the Sectoral Labour Market

The paper describes the application of the Hamiltonian System in deriving a solution to the dynamic system of the sectoral labour market. The aim is to determine the best control that would allow maximum sector participation in a three-sector labour market. The three sectors of the labour market are the goods-producing sector, the service-providing sector, and the agriculture sector. The sectoral labour market is formulated as an optimal control problem whose solution is sought using the Hamiltonian system by following the necessary conditions for optimality in the Hamiltonian System of Pontryagin’s Maximum Principle. The paper considered that the optimal control would be the measure of labour market efforts to produce the general worker’s population, and any split would mean there is production of active sector productive workers. The sector with the highest optimal allocation of effort, defined as the control within a fixed time, is identified as the best-performing sector. Service-providing sector, in this case, presented the best control. The obtained optimal control parameter is later applied to the experimental data to check on the effect of the control on the sectoral labour market participation rate. The results of the problem indicated that the application of the control had a significant effect on the sectoral labour market participation rate as compared to when there was no control. The sector with the highest annual growth rate measured as the sectoral participation rate was identified as the goods-producing sector

Optimal Trajectories of a UAV Base Station Using Hamilton-Jacobi Equations

We consider the problem of optimizing the trajectory of an Unmanned Aerial Vehicle (UAV). Assuming a traffic intensity map of users to be served, the UAV must travel from a given initial location to a final position within a given duration and serves the traffic on its way. The problem consists in finding the optimal trajectory that minimizes a certain cost depending on the velocity and on the amount of served traffic. We formulate the problem using the framework of Lagrangian mechanics. We derive closed-form formulas for the optimal trajectory when the traffic intensity is quadratic (single-phase) using Hamilton-Jacobi equations. When the traffic intensity is bi-phase, i.e. made of two quadratics, we provide necessary conditions of optimality that allow us to propose a gradient-based algorithm and a new algorithm based on the linear control properties of the quadratic model. These two solutions are of very low complexity because they rely on fast convergence numerical schemes and closed form formulas. These two approaches return a trajectory satisfying the necessary conditions of optimality. At last, we propose a data processing procedure based on a modified K-means algorithm to derive a bi-phase model and an optimal trajectory simulation from real traffic data.

A fractional-order mathematical model for lung cancer incorporating integrated therapeutic approaches

This paper addresses the dynamics of lung cancer by employing a fractional-order mathematical model that investigates the combined therapy of surgery and immunotherapy. The significance of this study lies in its exploration of the effects of surgery and immunotherapy on tumor growth rate and the immune response to cancer cells. To optimize the treatment dosage based on tumor response, a feedback control system is designed using control theory, and Pontryagin’s Maximum Principle is utilized to derive the necessary conditions for optimality. The results reveal that the reproduction number (R_0) is 2.6, indicating that a lung cancer cell would generate 2.6 new cancer cells during its lifetime. The reproduction coefficient (R_c) is 0.22, signifying that cancer cells divide at a rate that is 0.22 times that of normal cells. The simulations demonstrate that the combined therapy approach yields significantly improved patient outcomes compared to either treatment alone. Furthermore, the analysis highlights the sensitivity of the steady-state solution to variations in k_5 (the rate of division of cancer stem cells) and k_{13} (the rate of differentiation of cancer stem cells into progenitor cells). This research offers clinicians a valuable tool for developing personalized treatment plans for lung cancer patients, incorporating individual patient factors and tumor characteristics. The novelty of this work lies in its integration of surgery, immunotherapy, and control theory, extending beyond previous efforts in the literature.

Testing Lennard-Jones clusters for optimality

This note advertises a simple necessary condition for optimality that any list N ↦ vx(N) of computer-generated putative lowest average pair energies vx(N) of clusters that consist of N monomers has to satisfy whenever the monomers interact with each other through pair forces satisfying Newton’s “action equals re-action.” These can be quite complicated, as, for instance, in the TIP5P model with five-site potential for a rigid tetrahedral-shaped H2O monomer of water, or as simple as the Lennard-Jones single-site potential for the center of an atomic monomer (which is also used for one site of the H2O monomer in the TIP5P model, which in addition has four peripheral sites with Coulomb potentials). The empirical usefulness of the necessary condition is demonstrated by testing a list of publicly available Lennard-Jones cluster data that have been pooled from 17 sources, covering the interval 2 ≤ N ≤ 1610 without gaps. The data point for N = 447 failed this test, meaning the listed 447-particle Lennard-Jones cluster energy was not optimal. To implement this test for optimality in search algorithms for putatively optimal configurations is an easy task. Publishing only the data that pass the test would increase the odds that these are actually optimal, without guaranteeing it, though.

Classical Continuous Constraint Boundary Optimal Control Vector Problem for Triple Nonlinear Parabolic System

In this paper, our purpose is to study the classical continuous constraints boundary optimal triple control vector problem dominating nonlinear triple parabolic boundary value problem. The existence theorem for a classical continuous triple optimal control vector CCCBOTCV is stated and proved under suitable assumptions. The mathematical formulation of the adjoint triple boundary value problem associated with the nonlinear triple parabolic boundary value problem is discovered. The Fréchet derivative of the Hamiltonian is derived. Under proper assumptions, both theorems are granted; the necessary conditions for optimality and the sufficient conditions for optimality of the classical continuous constraints boundary optimal triple control vector problem are stated and proven.

Optimal feedback control law for automated vehicles in the presence of cyberattacks: A min–max approach

While automated vehicles (AVs) are expected to bring a wide range of benefits to future transportation systems, emerging AV technologies open a door for cyberattacks, where a select number of AVs are compromised to drive in an adversarial manner. This could result in network-wide increases in traffic congestion and energy consumption, degrading the performance of transportation systems. Hence, it is necessary to develop attack mitigation strategies for AVs as AVs gradually become a reality. However, this is rather challenging due to the lack of knowledge of adversaries. Limited prior studies have assumed attacks with a given probability distribution without considering their malicious intention, which is far from realistic due to the adversarial nature of potential attacks.In this study, we derive an optimal feedback control law for AVs in the presence of cyberattacks. Notably, attacks are only assumed to have a bounded magnitude without being subject to any specific statistical distribution, which is not only of theoretical interest but also relaxes the assumptions seen in prior studies. More importantly, to deal with lack of knowledge of malicious attacks, we, for the first time, formulate a min–max control problem to minimize the worst-case disturbance to traffic flow. Specifically, we present a mathematical framework for describing mixed-autonomy traffic involving AVs and human-driven vehicles (HVs), considering two typical types of attacks, namely false data injection attack on sensor measurements and malicious attack on AV control commands. Based on the framework presented, we derive a set of necessary conditions of optimality for the min–max control problem, based on which an iterative computational algorithm is developed for determining the optimal control (driving) strategy of AVs. The effectiveness of the proposed approach is demonstrated via extensive numerical simulations.