A class of finite time horizon optimal control problems with nonlinear dynamics and non-quadratic costs is considered. Stat-quad duality is used to transform the problem into a canonical form. A derivative-free numerical method that only uses fixed-point iterations is devised to solve it efficiently, the convergence of which is limited only by the existence of the staticizing control process (argstat). For problems with mild and low-dimensional nonlinearities, this leads to dimension reduction of the control space. A 4-D and a 25-D control problem are solved to demonstrate its accuracy and scalability.

With a view to generalizing existing max-plus and min-plus methods so as to use quadratic basis functions with a non-uniform Hessian, the underlying dual space representation of dynamic programming founded on the semiconvex transform is generalized via the quadratic transform. Using this generalization, an illustrative iteration is proposed as the foundation of a new max-plus method for the solution of a class of optimal control problems.

In coverage control problems that involve time-varying density functions, the coverage control law depends on spatial integrals of the time evolution of the density function. The latter is often neglected, replaced with an upper bound or calculated as a numerical approximation of the spatial integrals involved. In this paper, we consider a special case of time-varying density functions modeled as Gaussian Mixture Models (GMMs) that evolve with time via a set of time-varying sources (with known corresponding velocities). By imposing this structure, we obtain an efficient time-varying coverage controller that fully incorporates the time evolution of the density function. We show that the induced trajectories under our control law minimise the overall coverage cost. We elicit the structure of the proposed controller and compare it with a classical time-varying coverage controller, against which we benchmark the coverage performance in simulation. Furthermore, we highlight that the computationally efficient and distributed nature of the proposed control law makes it ideal for multi-vehicle robotic applications involving time-varying coverage control problems. We employ our method in plume monitoring using a swarm of drones. In an experimental field trial we show that drones guided by the proposed controller are able to track a simulated time-varying chemical plume in a distributed manner.

A typical coverage control objective function, known as the locational cost, is ill-suited for theoretical analysis of time-varying coverage control. This paper proposes a family of objective functions that unify and generalise several objective functions from the literature. Coverage control laws that enforce a decrease condition on an objective function from this family render the multi-agent system locally stable and optimal with respect to the locational cost in the time-varying coverage control setting. Two distinct classes of time-varying coverage controller are proposed: one that tracks centroids via inversion of local position-centroid kinematics, and another that directly descends the objective function, obviating computationally expensive matrix inversions. Local stability of both coverage controllers with time-varying density functions and coverage regions is established using the proposed objective functions via a Lyapunov-style argument. Simulation results demonstrate improved centroid-tracking performance when compared to existing coverage controllers in the literature.

Multifidelity models integrate data from multiple sources to produce a single approximator for the underlying process. Dense low-fidelity samples are used to reduce interpolation error, while sparse high-fidelity samples are used to compensate for bias or noise in the low-fidelity samples. Deep Gaussian processes (GPs) are attractive for multifidelity modelling as they are non-parametric, robust to overfitting, perform well for small datasets, and, critically, can capture nonlinear and input-dependent relationships between data of different fidelities. Many datasets naturally contain gradient data, especially when they are generated by computational models that are compatible with automatic differentiation or have adjoint solutions. Principally, this work extends deep GPs to incorporate gradient data. We demonstrate this method on an analytical test problem and a realistic partial differential equation problem, where we predict the aerodynamic coefficients of a hypersonic flight vehicle over a range of flight conditions and geometries. In both examples, the gradient-enhanced deep GP outperforms a gradient-enhanced linear GP model and their non-gradient-enhanced counterparts.

Reachable sets for a dynamical system describe collections of system states that can be reached in finite time, subject to system dynamics. They can be used to guarantee goal satisfaction in controller design or to verify that unsafe regions will be avoided. However, general-purpose methods for computing these sets suffer from the curse of dimensionality, which typically prohibits their use for systems with more than a small number of states, even if they are linear. In this paper, we demonstrate that viscosity supersolutions and subsolutions of a Hamilton-Jacobi-Bellman equation can be used to generate, respectively, under-approximating and over-approximating reachable sets for time-varying nonlinear systems. Based on this observation, we derive dynamics for a union and intersection of ellipsoidal sets that, respectively, under-approximate and over-approximate the reachable set for linear time-varying systems subject to an ellipsoidal input constraint and an ellipsoidal terminal (or initial) set. We demonstrate that the dynamics for these ellipsoids can be selected to ensure that their boundaries coincide with the boundary of the exact reachable set along a solution of the system. The ellipsoidal sets can be generated with polynomial computational complexity in the number of states, making our approximation scheme computationally tractable for continuous-time linear time-varying systems of relatively high dimension.

This paper develops an algorithm for upper- and lower-bounding the value function for a class of linear time-varying games subject to convex control sets. In particular, a two-player zero-sum differential game is considered where the respective players aim to minimise and maximise a convex terminal state cost. A collection of solutions of a single-player dynamical system subject to a trimmed control set is used to characterise a viscosity supersolution of a Hamilton-Jacobi (HJ) equation, which in turn yields an upper bound for the value function. Analogously, a collection of hyperplanes is used to characterise a viscosity subsolution of the HJ equation, which yields a lower bound. The computational complexity and memory requirement of the proposed algorithm scales with the number of solutions and hyperplanes that characterise the bounds, which is not explicitly tied to the number of system states. Thus, the algorithm is tractable for systems of moderately high dimension whilst preserving rigorous guarantees for optimal control and differential game applications.

A finite-horizon nonlinear optimal control problem is considered. Stat-quad duality is used to generate an equivalent problem with linear dynamics and a modification term in the running cost and two auxiliary controls processes. This problem form is used to obtain a representation of the value function as a staticization problem over a set of quadratic functions, where the coefficients of the quadratics consists of the solution to a differential Riccati equation, a linear ODE and an integral. This representation allows the value function to be evaluated independently at any time and any point in the state space. A specialized numerical method is proposed for solving the resulting staticization problem, which is able to leverage the low dimensionality of nonlinearity. A numerical example with five-dimensional state space is included.

This paper proposes a distributed control law for the coverage of time-varying density functions and coverage regions with dynamics that are integrable. Asymptotic stability guarantees are established via a variation of Barbalat’s Lemma. In doing so we propose an objective function that accounts for the explicit time dynamics associated with the local minima of a standard objective function used in the coverage control literature. This work advances upon existing literature on time-varying coverage control by providing stability guarantees in the distributed case without requiring unbounded control inputs. These guarantees are then validated through simulation, where the performance of a proposed controller is then compared to an existing feedback control law that is well known in the literature.

This paper develops an algorithm for upper-bounding the value function for a class of continuous-time optimal control problems. The upper bound can be used as a conservative estimate for the minimum cost that can be attained by any constraint admissible control from some initial state. Linear time-varying systems subject to convex input constraints and a state-independent running cost are considered. A collection of solutions of an augmented dynamical system is used to characterise viscosity supersolutions of a Hamilton-Jacobi-Bellman equation, which in turn yields an upper bound for the value function. The proposed algorithm has a computational complexity that scales in the number of these solutions as opposed to the dimension of the system, making the algorithm tractable for high dimensional systems.

A finite-horizon nonlinear optimal control problem is considered. Stat-quad duality is used to generate an equivalent problem with linear dynamics and running cost that is quadratic in state with an additional term that is nonlinear in newly introduced control state variables. The new problem form is used to obtain a representation of the value function in terms of staticization over a set of quadratic functions, where the coefficients of the quadratic functions consist of the solutions to certain ODEs. A novel numerical method is indicated for solution of the resulting staticization problem; the method leverages the low dimensionality of nonlinearity. An example is included. Copyright (c) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

In this paper feedback laws for a class of infinite horizon control problems under state constraints are investigated. We provide a two-player game representation for such control problems assuming time dependent dynamics and Lagrangian and the set constraints merely compact. Using viability results recently investigated for state constrained problems in an infinite horizon setting, we extend some known results for the linear quadratic regulator problem to a class of control problems with nonlinear dynamics in the state and affine in the control. Feedback laws are obtained under suitable controllability assumptions.

The coverage control problem involves spatially disseminating a network of mobile agents using distributed control laws, or coverage controllers, over a desired region to locally minimize an associated cost function. Mobile robots using coverage controllers will be subject to various disturbances, such as uncertainty, errors, and delays. Asymptotic stability of [Cortes et al.'s 2004] coverage controller and its variations have been studied; however, rates of convergence are omitted. Recent work has provided convergence rates for a coverage controller under certain assumptions, including regions where the network locally converges exponentially. The main contribution of this work is to show that a variation of [Cortes et al.'s 2004] coverage controller also features robust stability properties, specifically input-to-output stability, under the same set of assumptions. Conservative bounds are used to provide theoretical guarantees on stability, and simulations are used to verify the results and highlight practical performance of the controller.

Although numerical schemes exist for approximating the value of an optimal control problem, the curse-of-dimensionality limits their application in practice. In this paper, super- and subsolutions of a Hamilton-Jacobi equation are used to characterise upper and lower bounds of a corresponding value function that hold locally over sublevel, superlevel, or 'thick' level sets of the bounding functions. In reachability analysis, these bounds may facilitate the tractable computation of inner and outer approximations of reachable sets. The value of Mayer problems for time-varying, continuous-time nonlinear systems with input constraints are considered. Copyright (c) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

Reachability analysis is a powerful tool when it comes to capturing the behaviour, thus verifying the safety, of autonomous systems. However, general-purpose methods, such as Hamilton-Jacobi approaches, exhibit exponential computational complexity with respect to the state dimension. In this paper, we show that supersolutions and subsolutions of a Hamilton-Jacobi-Bellman equation can be used to generate under- and over-approximating reachable sets for nonlinear systems, and based on this, we develop a scheme for approximating reachable sets of linear time-invariant systems via ellipsoids with polynomial computational complexity.

Controllability and feasiblity measures are used to determine whether a given system can achieve its specified objective. However, for nonlinear systems with state constraints, the controllable and feasible sets may be highly sensitive to minor perturbations in the system's constraints, initial states and parameters. This becomes particularly important in codesign of hypersonic vehicles, where functions governing the dynamics must be estimated from expensive computational fluid dynamics simulations, and poor initialization can lead to significant waste of resources. By relaxation of the constraints and introduction of a surrogate cost, we provide a method for detecting and quantifying which constraints are violated. To demonstrate the method in a concrete example, we apply the technique to simulation of hypersonic vehicle trajectories.

Distribution of a network of mobile agents over a given region, subject to various environmental factors, is desirable for a variety of applications. This problem is referred to as the coverage control problem in the motion-coordination literature, and has seen many variations and augmentations to enhance the network's coverage capabilities. Certain nonlinear controllers derived using Lyapunov theory feature desirable properties, such as distributed communication and computation. This work demonstrates that a variation of Cortés et al.’s coverage controller is state-independent input-to-output stable under a series of assumptions. The stability property is validated through experimentation on a hardware platform with a variety of disturbances present in real-world systems, such as estimation errors and propagation delay.

A class of nonlinear, stochastic staticization control problems (including minimization problems with smooth, convex, coercive payoffs) driven by diffusion dynamics with constant diffusion coefficient is considered. The nonlinearities are addressed through stat duality. The second-order Hamilton-Jacobi partial differential equation (HJ PDE) is converted into a first-order HJ PDE in the dual variable, which, however, contains a correction term. Approximations to the correction term are indicated. A numerical example is included.