Bellman Function Research Articles

Optimized Discrete Nonlinear Control of Alternating Current Three-Phase Motors via an Industrial Variable Frequency Drive

This article presents a suboptimal nonlinear control strategy to improve the dynamics of a three-phase alternating current (AC) motor. Using dynamic programming, the calculation of the Bellman function is avoided by determining a suboptimal control sequence that locally minimizes a quadratic performance index at each step. The motor’s fixed-frame nonlinear mathematical model controls the stator currents, rotor magnetic fluxes, and rotor angular speed by applying voltages to the stator. Experimental tests are conducted using a Delta VFD007EL11A variable frequency drive (VFD), demonstrating improved motor state behavior and performance compared to an optimal proportional integral (PI) control and a fixed reference input in the VFD. The experiments include set point changes and a comparative analysis of the energy consumption between both controllers considering two cases: free and with load on the motor shaft.

On Maz’ya’s Φ-Inequalities for Martingale Fractional Integration and Their Bellman Functions

Inspired by a conjecture of Vladimir Maz’ya on Φ-inequalities in the spirit of Bourgain and Brezis, we establish some Φ-inequalities for fractional martingale transforms. These inequalities may be thought of as martingale models of Φ-inequalities for differential operators. The proofs rest on new simple Bellman functions.

Active Fault Detection Based on Tensor Train Decomposition

The paper deals with active fault detection of stochastic systems based on tensor train representation of the Bellman function. The faulty and faulty-free behavior of the system is represented using multiple models. The active fault detection problem is treated as an optimal design problem similar to optimal stochastic control. The original problem is reformulated as a perfect state information problem by introducing an information state that contains statistics computed by a state estimator. The Bellman function is computed using the value iteration algorithm over a rectilinear grid set up in the information state space. Within the value iteration algorithm, the Bellman function is represented using the tensor train decomposition, and considerable attention is devoted to designing a rectilinear grid that respects the constraints placed on the elements of the information state.

Formation control scheme with reinforcement learning strategy for a group of multiple surface vehicles

AbstractThis article presents a comprehensive approach to integrate formation tracking control and optimal control for a fleet of multiple surface vehicles (SVs), accounting for both kinematic and dynamic models of each SV agent. The proposed control framework comprises two core components: a high‐level displacement‐based formation controller and a low‐level reinforcement learning (RL)‐based optimal control strategy for individual SV agents. The high‐level formation control law, employing a modified gradient method, is introduced to guide the SVs in achieving desired formations. Meanwhile, the low‐level control structure, featuring time‐varying references, incorporates the RL algorithm by transforming the time‐varying closed agent system into an equivalent autonomous system. The application of Lyapunov's direct approach, along with the existence of the Bellman function, guarantees the stability and optimality of the proposed design. Through extensive numerical simulations, encompassing various comparisons and scenarios, this study demonstrates the efficacy of the novel formation control strategy for multiple SV agent systems, showcasing its potential for real‐world applications.

Communication-Free Voltage-Regulation and Current-Sharing for DC Microgrids: An Intelligent Edge Control

Though voltage-regulation and current-sharing of distributed generations (DGs) in dc-microgrids have been widely studied, additional communication links or independent modulation circuit should be added to achieve information transmission. To accomplish precise current-sharing/voltage-regulation without additional communication devices, this article proposes a communication-free intelligent edge control regarding voltage and current for dc-microgrids. First, the power-information dual modulation (PIDM) is designed to achieve information exchange among DGs and eliminate additional communication devices. Second, the cooperative control problem with two coupled targets, i.e., accurate voltage-regulation and current-sharing, is converted into a matter of optimal control. Therefore, the voltage-regulation and current-sharing could be solved concurrently. In addition, the control objective function of each DG is switched to provide the optimal controller and minimize the voltage/current control deviation, which is further switched to solve the Hamilton–Jacobi–Bellman (HJB) function. In order to solve this HJB function, which is difficult to obtain analytical solution, an intelligent edge control strategy with PIDM is proposed to solve the HJB function. Therefore, the precise voltage-regulation and current-sharing can be accomplished. Finally, the proposed control approach is verified through simulation results.

Trilinear embedding for divergence-form operators with complex coefficients

We prove a dimension-free Lp(Ω)×Lq(Ω)×Lr(Ω)→L1(Ω×(0,∞)) embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on Ω, and for triples of exponents p,q,r∈(1,∞) mutually related by the identity 1/p+1/q+1/r=1. Here Ω is allowed to be an arbitrary open subset of Rd. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as p-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato–Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.

Dimension-free Lp-estimates for vectors of Riesz transforms in the rational Dunkl setting

In this article, we prove a dimension-free upper bound for the Lp-norms of the vector of Riesz transforms in the rational Dunkl setting. Our main technique is the Bellman function method adapted to the Dunkl setting.

Bilinear embedding in Orlicz spaces for divergence-form operators with complex coefficients

We prove a bi-sublinear embedding for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form and for certain mutually dual pairs of Orlicz-space norms. This generalizes a result by Carbonaro and Dragičević from power functions to more general Young functions that still behave like powers. To achieve this, we generalize a Bellman function constructed by Nazarov and Treil.

A Guaranteed Deterministic Approach to Superhedging: The Relationship between the Deterministic and Probabilistic Problem Statements without Trading Constraints

We consider a guaranteed deterministic approach to the discrete-time superreplication problem in which it is required to cover a contingent liability on a written option in all feasible scenarios. These scenarios are described by a priori given compact sets depending on the price history: at each time instant, the price increments must lie in the corresponding compact sets. We assume no transaction costs. The problem statement is game-theoretic and leads to the Bellman--Isaacs equations in pure and mixed “market” strategies. In the case of no trading constraints, we study the relationship between the Bellman functions in the “deterministic” and “probabilistic” statements of the superhedging problem. As established under very general conditions, the “probabilistic” Bellman function does not exceed the “deterministic” counterpart. Sufficient conditions for their coincidence are found.

Weighted estimates for the multilinear maximal operator

The paper contains the study of the weighted Lp1×Lp2×…×Lpm→Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p_1}\ imes L^{p_2}\ imes \\ldots \ imes L^{p_m}\\rightarrow L^p$$\\end{document} estimates for the multilinear maximal operator, in the context of abstract probability spaces equipped with a tree-like structure. Using the Bellman function method, we identify the associated optimal constants in the symmetric case p1=p2=…=pm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_1=p_2=\\ldots =p_m$$\\end{document}, and a tight constant for remaining choices of exponents.

Burkholder meets Gundy: Bellman function method for general operators on martingales

It is shown that the Bellman function method can be applied to study the Lp-norms of general operators on martingales, i.e., of operators that are not necessarily martingale transforms. Informally, we provide a single Bellman-type function that “encodes” the Lp-boundedness of “almost all” operators from Gundy's extrapolation theorem. As examples of such operators, we consider the Haar transforms and the operator whose Lp-boundedness underlies Rubio de Francia's inequality for the Walsh system.

On the best constant in the Lp estimate for the sharp maximal function

Let I be a fixed subinterval of R and let 1≤p<∞ be a fixed exponent. Suppose that f∈Lp(I) is an arbitrary function of integral zero and let f# be the BLO-based sharp maximal function of f. The paper contains the identification of the best constant Cp in the estimate‖f‖Lp≤Cp‖f#‖Lp. The proof rests on the explicit evaluation of the Bellman function associated with the above estimate.

Constrained Optimal Control for Nonlinear Multi-Input Safety-Critical Systems with Time-Varying Safety Constraints

In this paper, we investigate the constrained optimal control problem of nonlinear multi-input safety-critical systems with uncertain disturbances and time-varying safety constraints. By utilizing a barrier function transformation, together with a new disturbance-related term and a smooth safety boundary function, a nominal system-dependent multi-input barrier transformation architecture is developed to deal with the time-varying safety constraints and uncertain disturbances. Based on the obtained transformation system, the coupled Hamilton–Jacobi–Bellman (HJB) function is established to obtain the constrained Nash equilibrium solution. In addition, due to the fact that it is difficult to solve the HJB function directly, the single critic neural network (NN) is constructed to approximate the optimal performance index function of different control inputs, respectively. It is proved theoretically that, under the influence of uncertain disturbances and time-varying safety constraints, the system states and neural network parameters can be uniformly ultimately bounded (UUB) by the proposed neural network approximation method. Finally, the effectiveness of the proposed method is verified by two nonlinear simulation examples.

On a Bellman function associated with the Chang–Wilson–Wolff theorem: a case study

The tail of distribution (i.e., the measure of the set { f ≥ x } \{f\ge x\} ) is estimated for those functions f f whose dyadic square function is bounded by a given constant. In particular, an estimate following from the Chang–Wilson–Wolf theorem is slightly improved. The study of the Bellman function corresponding to the problem reveals a curious structure of this function: it has jumps of the first derivative at a dense subset of the interval [ 0 , 1 ] [0,1] (where it is calculated exactly), but it is of C ∞ C^\infty -class for x &gt; 3 x&gt;\sqrt 3 (where it is calculated up to a multiplicative constant). An unusual feature of the paper consists of the usage of computer calculations in the proof. Nevertheless, all the proofs are quite rigorous, since only the integer arithmetic was assigned to a computer.

Sharp moment estimates for martingales with uniformly bounded square functions

We provide sharp bounds for the exponential moments and $p$-moments, $1\leqslant p \leqslant 2$, of the terminate distribution of a martingale whose square function is uniformly bounded by one. We introduce a Bellman function for the corresponding extremal problem and reduce it to the already known Bellman function on $\mathrm{BMO}([0,1])$. In the case of tail estimates, a similar reduction does not work exactly, so we come up with a fine supersolution that leads to sharp tail estimates.

Sharp multiplicative inequalities with BMO II

We find the best possible constant C in the inequality‖φ‖Lrpr⩽C‖φ‖Lppr‖φ‖BMO1−pr for all values of parameters p and r such that 1⩽p<r<+∞. To solve the problem, we define and explicitly compute the corresponding Bellman function. This function depends on three variables and has a rather complicated structure. We first solve the problem on an interval and then transfer our results to the circle and the line. We also obtain explicit estimates in several higher-dimensional settings.

The Bellman function of the dyadic maximal operator in connection with the dyadic Carleson imbedding theorem

Abstract We provide an alternative proof and expression of the Bellman function of the dyadic maximal operator in connection with the dyadic Carleson imbedding theorem, which appears in [A. D. Melas, The Bellman functions of dyadic-like maximal operators and related inequalities, Adv. Math. 192 2005, 2, 310–340]. We also evaluate the Bellman function of four variables of the dyadic maximal operator.

Book Review: The Bellman function technique in harmonic analysis

Book Review: The Bellman function technique in harmonic analysis

Self‐learning‐based optimal tracking control of an unmanned surface vehicle with pose and velocity constraints

AbstractIn this article, subject to both pose and velocity constraints within narrow waters, a self‐learning‐based optimal tracking control (SLOTC) scheme is innovatively created for an unmanned surface vehicle (USV) by deploying actor‐critic reinforcement learning (RL) mechanism and backstepping technique. To be specific, the barrier Lyapunov function (BLF) is devised to uniformly limit the states within a predefined region pertaining to a smoothly feasible reference trajectory. By virtue of a constrained Hamilton–Jacobi–Bellman (HJB) function, an actor‐critic control structure under backstepping is established by employing adaptive neural network identifiers which recursively updates actor and critic, simultaneously. Eventually, theoretical analysis proves that the entire SLOTC scheme can render all the states remain in the predefined compact set while tracking errors converge to an arbitrarily small neighborhood of the origin. Simulation results on a prototype USV demonstrate remarkable effectiveness and superiority.

Sharp L_p estimates for paraproducts on general measure spaces

The paper contains the identification of the L_p norms of paraproducts, defined on general measure spaces equipped with a dyadic-like structure. The proof exploits the Bellman function method.