Thompson sampling (TS) is an effective method to explore parametric uncertainties and can therefore be used for active learning-based controller design. However, TS relies on finite parametric representations, which limits its applicability to more general spaces, which are more commonly encountered in control system design. To address this issue, this work pro poses a parameterization method for control law learning using reproducing kernel Hilbert spaces and designs a data-driven active learning control approach. Specifically, the proposed method treats the control law as an element in a function space, allowing the design of control laws without imposing restrictions on the system structure or the form of the controller. A TS framework is proposed in this work to explore potential optimal control laws, and the convergence guarantees are further provided for the learning process. Theoretical analysis shows that the proposed method learns the relationship between control laws and closed-loop performance metrics at an exponential rate, and the upper bound of control regret is also derived. Numerical experiments on controlling unknown nonlinear systems validate the effectiveness of the proposed method.

Our friend and mentor Peter Caines has, together with his colleagues, created new foundations for studying collective dynamics in complex systems. Of particular inspiration to us has been his pioneering work in mean-field games (MFGs) launched two decades ago [10, 24, 25], and the related field of mean-field control. Peter pointed the way to both formulate and solve the problem of collective dynamics arising in a large population of heterogeneous dynamical systems. In this paper we survey some elements of MFGs within the context of controlled coupled oscillators. We begin by introducing a model for a single oscillator: dθ(t) = (ω + u(t)) dt + σ dξ(t), mod 2π (1) where θ(t) [0, 2π) is the phase of the oscillator at time t, ω is the nominal frequency with units of radiansper-second, {ξ(t): t 0} is a standard Wiener process, and u(t) is a control signal whose interpretation depends on the context. Unless otherwise noted, the SDEs are interpreted in their Itô form.

This paper proposes a fully data-driven approach for optimal control of nonlinear control-affine systems represented by a stochastic diffusion. The focus is on the scenario where both the nonlinear dynamics and stage cost functions are unknown, while only control penalty function and constraints are provided. Leveraging the theory of reproducing kernel Hilbert spaces, we introduce novel kernel mean embeddings (KMEs) to identify the Markov transition operators associated with controlled diffusion processes. The KME learning approach seamlessly integrates with modern convex operator-theoretic Hamilton-Jacobi-Bellman recursions. Thus, unlike traditional dynamic programming methods, our approach exploits the ``kernel trick'' to break the curse of dimensionality. We demonstrate the effectiveness of our method through numerical examples, highlighting its ability to solve a large class of nonlinear optimal control problems.

This study introduces an approach to obtain a neighboring extremal optimal control (NEOC) solution for a closed-loop optimal control problem, applicable to a wide array of nonlinear systems and not necessarily quadratic performance indices. The approach involves investigating the variation incurred in the functional form of a known closed-loop optimal control law due to small, known parameter variations in the system equations or the performance index. The NEOC solution can formally be obtained by solving a linear partial differential equation, akin to those encountered in the iterative solution of a nonlinear Hamilton-Jacobi equation. Motivated by numerical procedures for solving these latter equations, we also propose a numerical algorithm based on the Galerkin algorithm, leveraging the use of basis functions to solve the underlying Hamilton-Jacobi equation of the original optimal control problem. The proposed approach simplifies the NEOC problem by reducing it to the solution of a simple set of linear equations, thereby eliminating the need for a full re-solution of the adjusted optimal control problem. Furthermore, the variation to the optimal performance index can be obtained as a function of both the system state and small changes in parameters, allowing the determination of the adjustment to an optimal control law given a small adjustment of parameters in the system or the performance index. Moreover, in order to handle large known parameter perturbations, we propose a homotopic approach that breaks down the single calculation of NEOC into a finite set of multiple steps. Finally, the validity of the claims and theory is supported by theoretical analysis and numerical simulations.

We present a theory of compositionality in stochastic optimal control, showing how task-optimal controllers can be constructed from certain primitives. The primitives are themselves feedback controllers pursuing their own agendas. They are mixed in proportion to how much progress they are making towards their agendas and how compatible their agendas are with the present task. The resulting composite control law is provably optimal when the problem belongs to a certain class. This class is rather general and yet has a number of unique properties - one of which is that the Bellman equation can be made linear even for non-linear or discrete dynamics.

In this paper, the disjunctive and conjunctive lattice piecewise affine (PWA) approximations of explicit linear model predictive control (MPC) are proposed. The training data are generated uniformly in the domain of interest, consisting of the state samples and corresponding affine control laws, based on which the lattice PWA approximations are constructed. Re-sampling of data is also proposed to guarantee that the lattice PWA approximations are identical to explicit MPC control law in the unique order (UO) regions containing the sample points as interior points. Additionally, under mild assumptions, the equivalence of the two lattice PWA approximations guarantees that the approximations are error-free in the domain of interest. The algorithms for deriving statistically error-free approximation to the explicit linear MPC are proposed and the complexity of the entire procedure is analyzed, which is polynomial with respect to the number of samples. The performance of the proposed approximation strategy is tested through two simulation examples, and the result shows that with a moderate number of sample points, we can construct lattice PWA approximations that are equivalent to optimal control law of the explicit linear MPC.

We present a theory of compositionality in stochastic optimal control, showing how task-optimal controllers can be constructed from certain primitives. The primitives are themselves feedback controllers pursuing their own agendas. They are mixed in proportion to how much progress they are making towards their agendas and how compatible their agendas are with the present task. The resulting composite control law is provably optimal when the problem belongs to a certain class. This class is rather general and yet has a number of unique properties - one of which is that the Bellman equation can be made linear even for non-linear or discrete dynamics.

We present a theory of compositionality in stochastic optimal control, showing how task-optimal controllers can be constructed from certain primitives. The primitives are themselves feedback controllers pursuing their own agendas. They are mixed in proportion to how much progress they are making towards their agendas and how compatible their agendas are with the present task. The resulting composite control law is provably optimal when the problem belongs to a certain class. This class is rather general and yet has a number of unique properties - one of which is that the Bellman equation can be made linear even for non-linear or discrete dynamics. This gives rise to the compositionality developed here. In the special case of linear dynamics and Gaussian noise our framework yields analytical solutions (i.e. non-linear mixtures of linear-quadratic regulators) without requiring the final cost to be quadratic. More generally, a natural set of control primitives can be constructed by applying SVD to Greens function of the Bellman equation. We illustrate the theory in the context of human arm movements. The ideas of optimality and compositionality are both very prominent in the field of motor control, yet they are hard to reconcile. Our work makes this possible.

Behavioral goals are achieved reliably and repeatedly with movements rarely reproducible in their detail. Here we offer an explanation: we show that not only are variability and goal achievement compatible, but indeed that allowing variability in redundant dimensions is the optimal control strategy in the face of uncertainty. The optimal feedback control laws for typical motor tasks obey a "minimal intervention" principle: deviations from the average trajectory are only corrected when they interfere with the task goals. The resulting behavior exhibits task-constrained variability, as well as synergetic coupling among actuators--which is another unexplained empirical phenomenon.

Behavioral goals are achieved reliably and repeatedly with movements rarely reproducible in their detail. Here we offer an explanation: we show that not only are variability and goal achievement compatible, but indeed that allowing variability in redundant dimensions is the optimal control strategy in the face of uncertainty. The optimal feedback control laws for typical motor tasks obey a "minimal intervention" principle: deviations from the average trajectory are only corrected when they interfere with the task goals. The resulting behavior exhibits task-constrained variability, as well as synergetic coupling among actuators--which is another unexplained empirical phenomenon.