Optimal Impulsive Control Research Articles

Robust Space Trajectory Design Using Belief Optimal Control

This paper presents a novel approach to the robust solution of optimal impulsive control problems under aleatory and epistemic uncertainty. The novel approach uses belief Markov decision processes to reformulate the control problem in terms of uncertainty distributions, called beliefs, rather than the realizations of the system states. This formulation leads to the definition of a belief optimal control problem where the cost function and constraints are functions of the uncertainty distributions. The control formulation encompasses orbit determination arcs as well. The belief optimization is solved with a shooting-like transcription and a nonlinear programming solver to optimize the resulting discretized problem. Both aleatory and epistemic uncertainties are propagated with a nonintrusive polynomial expansion to capture the nonlinearities of the dynamics. Finally, this new approach is applied to the robust optimization of a flyby trajectory of the Europa Clipper mission in a scenario characterized by knowledge, execution, and observation uncertainty.

Hybrid optimal impulse control

This paper is concerned with an optimal impulse control problem under a hybrid diffusion (or, regime switching) model, where the state of the system consists of a number of diffusions coupled by a continuous-time finite-state Markov chain. The objective is to minimize the expected discounted cost from exerting the impulse control in the infinite horizon. Based on the dynamic programming principle (DPP), the value function of the hybrid optimal impulse control problem is shown to be the unique viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation, which is in the form of a coupled system of variational inequalities. Moreover, a verification theorem as the sufficient condition for optimality of a solution is also established. The optimal impulse control, indicates when and how it is optimal to intervene, is described by the obstacle part of the HJB equation. Finally, the general theoretical results are applied to an optimal cash management problem. The value function in closed-form and an explicit optimal policy are obtained, which exhibit clearly the effect of regime switching on the agent’s decision.

Event-Triggered Impulsive Optimal Control for Continuous-Time Dynamic Systems with Input Time-Delay

Time-delay is an inevitable factor in practice, which may affect the performance of optimal control. In this paper, the event-triggered impulsive optimal control for linear continuous-time dynamic systems is studied. The event-triggered impulsive optimal feedback controller with input time-delay is presented, where the impulsive instants are determined by some designed event-triggering function and condition depending on the state of the system. Some sufficient conditions are given for guaranteeing the exponential stability with the optimal controller. Moreover, the Zeno-behavior for the impulsive instants is excluded. Finally, an example with numerical simulation is given to verify the validity of the theoretical results.

Mathematical modeling and analysis of biological control strategy of aphid population

<abstract><p>To study the biological control strategy of aphids, in this paper we propose host-parasitoid-predator models for the interactions among aphids, parasitic wasps and aphidophagous Coccinellids incorporating impulsive releases of Coccinellids, and then study the long-term control and limited time optimal control of aphids by adjusting release amount and release timing of Coccinellids. For the long-term control, the existence and stability of the aphid-eradication periodic solution are investigated and threshold conditions about the release amount and release period to ensure the ultimate extinction of the aphid population are obtained. For the limited-time control, three different optimal impulsive control problems are studied. A time rescaling technique and an optimization algorithm based on gradient are applied, and the optimal release amounts and timings of natural enemies are gained. Our simulations indicate that in the limited-time control, the optimal selection of release timing should be given higher priority compared with the release amount.</p></abstract>

Optimal impulse control of West Nile virus

<abstract><p>We construct a West Nile virus epidemic model that includes the interaction between the bird hosts and mosquito vectors, mosquito life stages (eggs, larvae, adults), and the dynamics of both larvicide and adulticide. We derive the basic reproduction number for the epidemic as the spectral radius of the next generation matrix. We formulate two impulsive optimal control problems which seek to balance the cost of insecticide applications (both the timing and application level) with the benefit of (1) vector control: reducing the number of mosquitoes or (2) disease control: reducing the disease burden. We reformulate these impulsive optimal control problems as nonlinear optimization problems and derive associated necessary conditions for the optimal controls. Numerical simulations are used to address three questions: How does the control and its impact on the system vary with the objective type? Is it beneficial to optimize the treatment timing? How does the control and its impact on the population vary with the type of pesticide used?</p></abstract>

Duality in optimal impulse control

We consider the optimal impulse control of a dynamical system defined by a fixed uncontrolled flow. This problem is associated with two pairs of linear programs for which we prove the solvability and the absence of the duality gaps. Finally, we show how to retrieve the optimal control strategy from the solutions of those linear programs. The theoretical issues are illustrated by the meaningful example on the controlled epidemic model.

Finite-time stability and optimal impulsive control for age-structured HIV model with time-varying delay and Lévy noise

Finite-time stability and optimal impulsive control for age-structured HIV model with time-varying delay and Lévy noise

Optimal impulsive blood glucose control through multiple injections

Despite numerous advances in glycemic control based on the artificial pancreas, Multiple Daily Injections (MDI) are still the most common method in type 1 diabetes mellitus patients. In the present study, an optimal impulsive control approach is introduced to attain a safe range of blood glucose considering the daily life factors, including food intake and exercise. Depending on the food intake and exercise, the proposed impulsive control signal can determine the optimal exogenous insulin dosages per day. MDI program for type 1 diabetic patients due to several overeating conditions, with and without exercise, is a blood glucose control program. This program can conveniently help the patients to manage unwanted conditions such as missed insulin injections or meals. The recommended approach is investigated on the comprehensive glucose-insulin model of diabetic subjects, and it effectively determines the time and magnitude of the insulin injection dosages. Based on the simulation results, three times injection mode in terms of hypoglycemia and severe hyperglycemia risks is efficient and dosages times and magnitudes are well determined in different food intake scenarios by the proposed approach. This approach can significantly help physicians to regulate blood glucose in a sophisticated optimal, reliable route.

Impulsive optimal control for drug treatment of influenza A virus in the host with impulsive-axis equivalent model

This paper aims at investigating the drug optimal problem for influenza A virus treatment with two main dilemmas: i) reducing the number of collected data and ii) considering the closed-loop system as a case of negative control direction with minimum drug usage. For decreasing the number of data, the equivalent model is established by the structure of fuzzy-rules emulated network using only one state-variable “viral titer” on the impulsive axis. The impulsive optimal controller is designed without any knowledge of patients and drug dynamics. As a result, the proposed controller can be utilized for a wide class of patients who have different drug reputations.Furthermore, the closed-loop performance is analyzed by the main theorem and the additional lemma is formulated for a case of negative control direction.Finally, the proposed scheme is validated by comparison tests with the fixed-dose FDA and the neuro-optimal controller.

Monotone solutions for mean field games master equations: finite state space and optimal stopping

We present a new notion of solution for mean field games master equations. This notion allows us to work with solutions which are merely continuous. We first prove results of uniqueness and stability for such solutions. It turns out that this notion is helpful to characterize the value function of mean field games of optimal stopping or impulse control and this is the topic of the second half of this paper. The notion of solution we introduce is only useful in the monotone case. In this article we focus on the finite state space case.

No Infimum Gap and Normality in Optimal Impulsive Control Under State Constraints

In this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.

Chemical and biological control of parasite-borne disease Schistosomiasis: An impulsive optimal control approach

This article deals in a continuous Schistosomiasis transmission model involving hosts human and freshwater snail, and two free living Schistosoma larva forms Miracidia and Cercariae. Along with derivation of basic reproduction number and standard equilibrium analysis, Hopf bifurcation phenomenon is demonstrated analytically and numerically. Parameter sensitivity analysis of system solution has been used to identify significant parameters for the change of disease dynamics. Impulsive optimal control of the same system, adding combined biological and chemical control of snail population to minimize human infection is undertaken. Snail natural predator crayfish is used for biological control and molluscicide is used for chemical control, where both controls are released in water periodically for finite number of times. Mathematically, the continuous system is converted to a discrete-continuous hybrid control problem applying Pontryagin’s Maximum Principle. The optimum solution revealed the quantity of each control to be released at each time point. The control analysis identified parameters crucial to success of combined control.

Infinite horizon impulse control problem with jumps and continuous switching costs

PurposeThe purpose of this paper is to show the existence results for adapted solutions of infinite horizon doubly reflected backward stochastic differential equations with jumps. These results are applied to get the existence of an optimal impulse control strategy for an infinite horizon impulse control problem.Design/methodology/approachThe main methods used to achieve the objectives of this paper are the properties of the Snell envelope which reduce the problem of impulse control to the existence of a pair of right continuous left limited processes. Some numerical results are provided to show the main results.FindingsIn this paper, the authors found the existence of a couple of processes via the notion of doubly reflected backward stochastic differential equation to prove the existence of an optimal strategy which maximizes the expected profit of a firm in an infinite horizon problem with jumps.Originality/valueIn this paper, the authors found new tools in stochastic analysis. They extend to the infinite horizon case the results of doubly reflected backward stochastic differential equations with jumps. Then the authors prove the existence of processes using Envelope Snell to find an optimal strategy of our control problem.

Aggregated occupation measures and linear programming approach to constrained impulse control problems

For a constrained optimal impulse control problem of an abstract dynamical system, we introduce the occupation measures along with the aggregated occupation measures and present two associated linear programs. We prove that the two linear programs are equivalent under appropriate conditions, and each linear program gives rise to an optimal strategy in the original impulse control problem. In particular, we show the absence of the relaxation gap. By means of an example, we also present a detailed comparison of the occupation measures and linear programs introduced here with the related notions in the literature.

Study of the sterile insect release technique for a two-sex mosquito population model.

In this paper, to study the large-scale time control and limited-time control of mosquito population in a field, a two-sex mosquito population model with stage structure and impulsive releases of sterile males is proposed. For the large-scale time control, a wild mosquito-free periodic solution is given and conditions under which it is globally stable are obtained by the use of the monotone system theory. Besides, based on the stability analysis, threshold conditions under which the wild mosquito population is eliminated or not are obtained. Then we study three different optimal release strategies for the limited-time control, which takes into account both of the population control level of wild mosquitoes and the economic input. To solve technical problems in optimal impulsive control, a time rescaling technique is applied and the gradients of cost function with respect to all control parameters are obtained. In addition, by the aid of numerical simulation, we get the optimal release amounts and release timings for each release strategy. Our study indicates that the optimal release timing control is superior to the optimal release amount control. However, simultaneous optimal selection of release amount and release timing leads to the best control performance.

Оптимизация импульсных управляемых систем с промежуточными фазовыми ограничениями

In this paper, we consider an optimal impulsive control problem with intermediate state constraints. The peculiarity of the problem consists in a non-standard way of specifying of intermediate constraints. So the constraints must be satisfied for at least one selection of a set-valued solution to the impulsive control system. We prove a theorem for the existence of an optimal control and propose the reduction procedure that transforms the initial optimal control problem with intermediate constraints into a hybrid problem with control parameters. This hybrid problem gives an equivalent description of the optimal impulsive control problem. We discuss a numerical algorithm based on a direct collocation method and give a schema to the corresponding numerical calculations for a test example.

Optimal Impulse Control for Growth-Restricted Linear Diffusions with Regime Switching

This paper investigates the impulse cash-flow withdrawal (dividend distribution or consumption) optimization problem for a broad class of growth restricted linear diffusions with drift and volatility dependent on both the level of surplus and the environment/economy regime (described by a Markov chain). There are proportional and fixed costs associated with each cash-flow withdrawal transaction, and the objective of the optimization problem is to determine the optimal policy that maximizes the expected total cumulative discounted cash-flow withdrawals (dividend payments/consumption) minus expenses. This work extends all the existing work on the same problem under the diffusion setting by extending the underlying dynamics to a much more general class of Markov modulated diffusions and unifies the existing work on optimal impulse dividend control. We show that when there are two regimes, the value function, as a function of the initial cash reservoir and the initial environmental state, can be determined by solving a set of second order ordinary differential equations. We further show that either the optimal strategy is a regime switching lump-sum dividend barrier strategy or there exists no optimal strategies.

Continuity of the value function for deterministic optimal impulse control with terminal state constraint

Deterministic optimal impulse control problem with terminal state constraint is considered. Due to the appearance of the terminal state constraint, the value function might be discontinuous in general. The main contribution of this paper is the introduction of an intrinsic condition under which the value function is proved to be continuous. Then by a Bellman dynamic programming principle, the corresponding Hamilton-Jacobi-Bellman type quasi-variational inequality (QVI, for short) is derived. The value function is proved to be a viscosity solution to such a QVI. The issue of whether the value function is characterized as the unique viscosity solution to this QVI is carefully addressed and the answer is left open challengingly.

Optimizing Multiple Drug Administration from Depot by Applying Pharmacokinetic Concepts

For the multiple drug administration from therapeutic reasons it is important to maintain the concentration in the blood plasma in an appropriate range. In the present paper an optimization approach is developed to determine drug dosage regimen to achieve the desired plasma concentrations after application from depot, i.e. oral, muscular, subcutant. The developed methodology allows the optimization of both the dose and the dosage interval. Performance of the developed methodology is evaluated by computing bias and precision of the estimated trough and peak drug concentrations that are reached after dosage regimen determinations. This article focuses on an optimal impulsive control of compartment model to individualise dosage regimens of Amikacin in the context of extended dosage intervals. Amikacin is an aminoglycoside antibiotic used to treat various bacterial infections.

Numerical Algorithms For State-Linear Optimal Impulsive Control Problems Based On Feedback Necessary Optimality Conditions

We propose and compare three numeric algorithms for optimal control of state-linear impulsive systems. The algorithms rely on the standard transformation of impulsive control problems through the discontinuous time rescaling, and the so-called “feedback”, direct and dual, maximum principles. The feedback maximum principles are variational necessary optimality conditions operating with feedback controls, which are designed through the usual constructions of the Pontryagin’s Maximum Principle (PMP); though these optimality conditions are formulated completely in the formalism of PMP, they essentially strengthen it. All the algorithms are non-local in the sense that they are aimed at improving non-optimal extrema of PMP (local minima), and, therefore, show the potential of global optimization.