The optimal visiting problem is the optimization of a trajectory that has to touch or pass as close as possible to a collection of target points. The problem does not verify the dynamic programming principle, and it needs a specific formulation to keep track of the visited target points. In this paper, we introduce a hybrid approach by adding a discontinuous part of the trajectory switching between a group of discrete states related to the targets. Then, we show the well-posedness of the related Hamilton-Jacobi problem, by reformulating the optimal visiting as a collection of time-dependent optimal stopping problems.
In an optimal visiting problem, we want to control a trajectory that has to pass as close as possible to a collection of target points or regions. We introduce a hybrid control-based approach for the classic problem where the trajectory can switch between a group of discrete states related to the targets of the problem. The model is subsequently adapted to a mean-field framework to study viability and crowd fluxes to model a multitude of indistinguishable players.
Motivated by an optimal visiting problem, we study a switching mean-field game on a network, where both a decisional and a switching time-variable is at disposal of the agents for what concerns, respectively, the instant to decide and the instant to perform the switch. Every switch between the nodes of the network represents a switch from $0$ to $1$ of one component of the string $p = (p_1,\ldots, p_n)$ which, in the optimal visiting interpretation, gives information on the visited targets, being the targets labeled by $i=1,\ldots, n$. The goal is to reach the final string $(1, \ldots, 1)$ in …
We consider the optimal multi-agent persistent monitoring problem defined by a team of cooperating agents visiting a set of nodes (targets) on a graph with the objective of minimizing a measure of overall node state uncertainty. The solution to this problem involves agent trajectories defined both by the sequence of nodes to be visited by each agent and the amount of time spent at each node. Since such optimal trajectories are generally intractable, we propose a class of distributed threshold-based parametric controllers through which agent transitions from one node to the next are controlled …
In this research, a new variant of the vehicle routing problem with time windows is addressed. The nodes associated with the customers are related to each other through AND/OR precedence constraints. The objective is minimizing the total traveling and service time. This generalization is necessary for problems where the visiting node sequence is defined according to the AND/OR relations, such as picker routing problems. We propose a Mixed Integer Linear Programming model to solve small-scale instances extended from the well-known Solomon benchmark. A meta-heuristic algorithm based on the hybri…
Isfahan University of Technology（伊斯法罕理工大学）；Simon Fraser University（西蒙弗雷泽大学）
The Generalized Traveling Salesman Problem (GTSP) is an extension of the well-known Traveling Salesman Problem (TSP), where the node set is partitioned into clusters, and the objective is to find the shortest cycle visiting each cluster exactly once. In this paper, we present a new hybrid Ant Colony System (ACS) algorithm for the symmetric GTSP. The proposed algorithm is a modification of a simple ACS for the TSP improved by an efficient GTSP-specific local search procedure. Our extensive computational experiments show that the use of the local search procedure dramatically improves the perfor…
Among the most important variants of the traveling salesman problem (TSP) are those relaxing the constraint that every locus should necessarily get visited, rather taking into account a revenue (prize) for visiting customers. In the Profitable Tour Problem (PTP), we seek for a tour visiting a subset of customers while maximizing net gain (profit) as difference between total revenue collected from visited customers and incurred traveling costs. The metric TSP can be modeled as a PTP with large revenues. As such, PTP is well-known to be NP-hard and also APX-hardness follows. Nevertheless, PTP is…
This paper contains material originally presented in the first submission of the paper "On the Optimal Control of Impulsive Hybrid Systems On Riemannian Manifolds" to SIAM Journal on Control and Optimization on the 28th of February, 2012. In the revised version of the paper the following sections are removed.
We present a numerically tractable formulation for computing the optimal control of the class of hybrid dynamical systems whose trajectories are continuous. Our formulation, an extension of existing relaxed-control techniques for switched dynamical systems, incorporates the domain information of each discrete mode as part of the constraints in the optimization problem. Moreover, our numerical results are consistent with phenomena that are particular to hybrid systems, such as the creation of sliding trajectories between discrete modes.
This paper is concerned with optimal control problems for control systems in continuous time, and interacting particle system methods designed to construct approximate control solutions. Particular attention is given to the linear quadratic (LQ) control problem. There is a growing interest in re-visiting this classical problem, in part due to the successes of reinforcement learning (RL). The main question of this body of research (and also of our paper) is to approximate the optimal control law {\em without} explicitly solving the Riccati equation. A novel simulation-based algorithm, namely a …
A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. First- and second-order optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation of the time domains. For autonomous systems and time-independent integral cost, we prove that the Hamiltonian is constant in time when evaluated along the optimal controls and trajectories. A numerica…
This paper provides a geometrical derivation of the Hybrid Minimum Principle (HMP) for autonomous impulsive hybrid systems on Riemannian manifolds, i.e. systems where the manifold valued component of the hybrid state trajectory may have a jump discontinuity when the discrete component changes value. The analysis is expressed in terms of extremal trajectories on the cotangent bundle of the manifold state space. In the case of autonomous hybrid systems, switching manifolds are defined as smooth embedded submanifolds of the state manifold and the jump function is defined as a smooth map on the sw…
For a general optimal control problem for dynamical systems with hybrid dynamics, we study the dependency of the optimal cost and of the value function on the initial conditions, parameters, and perturbations. We show that upper and lower semicontinuous dependence of solutions on initial conditions -- properties that are captured by outer and inner well-posedness, respectively -- lead to the existence of a solution to the hybrid optimal control problem and upper/lower semicontinuity of the optimal cost. In particular, by exploiting properties of finite horizon reachable sets for hybrid systems…
A unique feature of hybrid dynamical systems (systems whose evolution is subject to both continuous- and discrete-time laws) is Zeno trajectories. Usually these trajectories are avoided as they can cause incorrect numerical results as the problem becomes ill-conditioned. However, these are difficult to justifiably avoid as determining when and where they occur is a non-trivial task. It turns out that in optimal control problems, not only can they not be avoided, but are sometimes required in synthesizing the solutions. This work explores the pedagogical example of the bouncing ball to demonstr…
We consider a nonlinear ordinary differential equation and want to control its behavior so that it reaches a target by minimizing a cost function. Our approach is to use hybrid systems to solve this problem: the complex dynamic is replaced by piecewise affine approximations which allow an analytical resolution. The sequence of affine models then forms a sequence of states of a hybrid automaton. Given a sequence of states, we introduce an hybrid approximation of the nonlinear controllable domain and propose a new algorithm computing a controllable, piecewise convex approximation. The same way t…
In this paper, we design a theoretical framework allowing to apply model predictive control on hybrid systems. For this, we develop a theory of approximate dynamic programming by leveraging the concept of alternating simulation. We show how to combine these notions in a branch and bound algorithm that can further refine the Q-functions using Lagrangian duality. We illustrate the approach on a numerical example.
The recent rise of electricity generation based on renewable energy sources increases the demand for transmission capacity. Capacity expansion via the upgrade of transmission line capacity, e.g., by conversion to a high-voltage direct current (HVDC) line, is an attractive option. In this paper, it is shown that if the upgrade to HVDC is applied systematically to selected transmission lines across the grid, a hybrid architecture is obtained that enables an efficient and globally optimal solution of the optimal power flow (OPF) problem. More precisely, for conventional meshed AC transmission gri…
This paper solves a path planning problem for a group of gliders. The gliders are tasked with visiting a set of interest points. The gliders have limited range but are able to increase their range by visiting special points called thermals. The problem addressed in this paper is of path planning for the gliders such that, the total number of interest points visited by the gliders is maximized. This is referred to as the multi-agent problem. The problem is solved by first decomposing it into several single-agent problems. In a single-agent problem a set of interest points are allocated to a sin…
Many steady-state problems in power systems, including rectangular power-voltage formulations of optimal power flows in the alternating-current model (ACOPF), can be cast as polynomial optimisation problems (POP). For a POP, one can derive strong convex relaxations, or rather hierarchies of ever stronger, but ever larger relaxations. We study means of switching from solving the convex relaxation to Newton method working on a non-convex Lagrangian of the POP.
A convex formulation is proposed for optimal energy management in aircraft with hybrid propulsion systems consisting of gas turbine and electric motor components. By combining a point-mass aircraft dynamical model with models of electrical and mechanical powertrain losses, the fuel consumed over a planned future flight path is minimised subject to constraints on the battery, electric motor and gas turbine. The resulting optimisation problem is used to define a predictive energy management control law that takes into account the variation in aircraft mass during flight. A simulation study based…
This paper considers the optimal control for hybrid systems whose trajectories transition between distinct subsystems when state-dependent constraints are satisfied. Though this class of systems is useful while modeling a variety of physical systems undergoing contact, the construction of a numerical method for their optimal control has proven challenging due to the combinatorial nature of the state-dependent switching and the potential discontinuities that arise during switches. This paper constructs a convex relaxation-based approach to solve this optimal control problem. Our approach begins…
This work focuses on optimal controls of a class of stochastic SIS epidemic models under regime switching. By assuming that a decision maker can either influence the infectivity period or isolate infected individuals, our aim is to minimize the expected discounted cost due to illness, medical treatment, and the adverse effect on the society. In addition, a model with the incorporation of vaccination is proposed. Numerical schemes are developed by approximating the continuous-time dynamics using Markov chain approximation methods. It is demonstrated that the approximation schemes converge to th…
Continued advances in PV and battery energy storage technologies have made hybrid PV-battery systems an attractive prospect for residential energy consumers. However the process to select an appropriate system is complicated by the relatively high cost of batteries, a multitude of available retail electricity plans and the removal of PV installation incentive schemes. In this paper, an optimization strategy based on an individual customer's temporal load profile is established to maximize electricity cost savings through optimal selection of PV-battery system size, orientation and retail elect…
This paper is concerned with a partially observed hybrid optimal control problem, where continuous dynamics and discrete events coexist and in particular, the continuous dynamics can be observed while the discrete events, described by a Markov chain, is not directly available. Such kind of problem is first considered in the literature and has wide applications in finance, management, engineering, and so on. There are three major contributions made in this paper: First, we develop a novel non-linear filtering method to convert the partially observed problem into a completely observed one. Our m…
In this paper we demonstrate a novel alternating direction method of multipliers (ADMM) algorithm for the solution of the hybrid vehicle energy management problem considering both power split and engine on/off decisions. The solution of a convex relaxation of the problem is used to initialize the optimization, which is necessarily nonconvex, and whilst only local convergence can be guaranteed, it is demonstrated that the algorithm will terminate with the optimal power split for the given engine switching sequence. The algorithm is compared in simulation against a charge-depleting/charge-sustai…
We study some optimal control problems on networks with junctions, approximate the junctions by a switching rule of delay-relay type and study the passage to the limit when $\varepsilon$, the parameter of the approximation, goes to zero. First, for a twofold junction problem we characterize the limit value function as viscosity solution and maximal subsolution of a suitable Hamilton-Jacobi problem. Then, for a threefold junction problem we consider two different approximations, recovering in both cases some uniqueness results in the sense of maximal subsolution.
This paper presents a nonlinear-model based hybrid optimal control technique to compute a suboptimal power-split strategy for power/energy management in a parallel hybrid electric vehicle (PHEV). The power-split strategy is obtained as model predictive control solution to the power management control problem (PMCP) of the PHEV, i.e., to decide upon the power distribution among the internal combustion engine, an electric drive, and other subsystems. A hierarchical control structure of the hybrid vehicle, i.e., supervisory level and local or subsystem level is assumed in this study. The PMCP con…
Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory is applied to these systems and a hybrid version of Pontryagin's maximum principle is presented. This hybrid maximum principle is presented to emphasize its geometric nature which makes its study amenable to the tools of geometric mechanics and symplectic geometry. One explicit benefit of this geometric approach is that Zeno behavior can be strongly controlled for "generic" control problems. Moreover, when the underlying control system is a mechanical impac…
Avanzi et al. (2016) recently studied an optimal dividend problem where dividends are paid both periodically and continuously with different transaction costs. In the Brownian model with Poissonian periodic dividend payment opportunities, they showed that the optimal strategy is either of the pure-continuous, pure-periodic, or hybrid-barrier type. In this paper, we generalize the results of their previous study to the dual (spectrally positive Lévy) model. The optimal strategy is again of the hybrid-barrier type and can be concisely expressed using the scale function. These results are confirm…
A well-know control policy in stochastic inventory control is the (R, s, S) policy, in which inventory is raised to an order-up-to-level S at a review instant R whenever it falls below reorder-level s. To date, little or no work has been devoted to developing approaches for computing (R, s, S) policy parameters. In this work, we introduce a hybrid approach that exploits tree search to compute optimal replenishment cycles, and stochastic dynamic programming to compute (s, S) levels for a given cycle. Up to 99.8% of the search tree is pruned by a branch-and-bound technique with bounds generated …
To contribute to a smart grid, the unit commitment and dispatch (UCD) is a very important problem to be revisited in the future microgrid environment. In this paper, the UCD problem is studied by taking account of both traditional thermal units and distributed generation units (DGs) while considering demand response (DR) and carbon emissions trading (CET). Then, the UCD problem is reformulated as the optimal switching problem of a hybrid system. The reformulation allows convenient handling of time-dependent start-up cost for dynamic programming (DP). Afterwards, a closed-loop hierarchical oper…
We consider the optimization of a dynamical system by switching at discrete time points between abstract evolution equations composed by nonlinearly perturbed strongly continuous semigroups, nonlinear state reset maps at mode transition times and Lagrange-type cost functions including switching costs. In particular, for a fixed sequence of modes, we derive necessary optimality conditions using an adjoint equation based representation for the gradient of the costs with respect to the switching times. For optimization with respect to the mode sequence, we discuss a mode-insertion gradient. The t…
We examine two configurations for parallel hybrid electric aircraft, one with, and one without, a mechanical connection between the engines and the electric motors. For this two designs, we then review the power allocation problem in the context of aircraft energy management for a 19-seat conceptual Hybrid Electric Aircraft. We then represent the original optimal control problem as a finite-dimensional optimization and validate the second-order sufficient conditions for global optimality of the obtained solution. This is then followed by a sensitivity analysis of the fuel consumption on the in…
In this note we propose a new variant of the hybrid variance-reduced proximal gradient method in [7] to solve a common stochastic composite nonconvex optimization problem under standard assumptions. We simply replace the independent unbiased estimator in our hybrid- SARAH estimator introduced in [7] by the stochastic gradient evaluated at the same sample, leading to the identical momentum-SARAH estimator introduced in [2]. This allows us to save one stochastic gradient per iteration compared to [7], and only requires two samples per iteration. Our algorithm is very simple and achieves optimal …
This paper studies the optimal investment problem for a hybrid pension plan under model uncertainty, where both the contribution and the benefit are adjusted depending on the performance of the plan. Furthermore, an age and time-dependent force of mortality and a linear maximum age are considered to capture the longevity trend. Suppose that the plan manager is ambiguity averse and is allowed to invest in a risk-free asset and a stock. The plan manager aims to find optimal investment strategies and optimal intergenerational risk-sharing arrangements by minimizing the cost of unstable contributi…
This paper concerns the numerical procedure for solving hybrid optimal control problems with sliding modes. A sliding mode is coped with differential-algebraic equations (DAEs) and that guarantees accurate tracking of the sliding motion surface. In the second part of the paper we demonstrate the correspondence between the discrete adjoint equations and the discretized version of the continuous adjoint equations in the case of system equations described by DAEs. We show that the discrete adjoint state trajectories converge to their continuous counterparts. Next, we describe the application of t…
This paper addresses modeling and control of a six-degree-of-freedom unmanned aerial vehicle capable of vertical take-off and landing in the presence of wind disturbances. We design a hybrid vehicle that combines the benefits of both the fixed-wing and the rotary-wing UAVs. A non-linear model for the hybrid vehicle is rapidly built, combining rigid body dynamics, aerodynamics of wing, and dynamics of the motor and propeller. Further, we design an H2 optimal controller to make the UAV robust to wind disturbances. We compare its results against that of PID and LQR-based control. Our proposed con…
This paper addresses the problem of planning successive Space Debris Collecting missions so that they can be achieved at minimal cost by a generic vehicle. The problem mixes combinatorial optimization to select and order the debris among a list of candidates, and continuous optimization to fix the rendezvous dates and to define the minimum fuel orbital maneuvers. The solution method proposed consists in three stages. Firstly the orbital transfer problem is simplified by considering a generic transfer strategy suited either to a high thrust or a low thrust vehicle. A response surface modelling …
