Boundedness Assumption Research Articles

A new Lagrangian-based first-order method for nonconvex constrained optimization

In this paper, we introduce a new form of Lagrangian function and propose a simple first-order primal-dual algorithm for solving nonconvex optimization with nonlinear equality constraints. We show that the algorithm generates bounded primal-dual iterates, and establish the convergence to KKT points under standard assumptions. The key features of the proposed method are: (i) it does not require boundedness assumptions on dual iterates generated by the algorithm as well as the set of multipliers; (ii) it is a single-loop algorithm that does not involve any penalty subproblems.

Liouville closed HT-fields

Let T be a complete, model complete o-minimal theory extending the theory of real closed ordered fields. An HT-field is a model K of T equipped with a T-derivation ▪ such that the underlying ordered differential field of ▪ is an H-field. We study HT-fields and their extensions. Our main result is that if T is power bounded, then every HT-field K has either exactly one or exactly two minimal Liouville closed HT-field extensions up to K-isomorphism. The assumption of power boundedness can be relaxed to allow for certain exponential cases, such as T=Th(Ran,exp).

Event-Triggered Near-Optimal Control for Unknown Discrete-Time Nonlinear Systems Using Parallel Control.

This article uses parallel control to investigate the problem of event-triggered near-optimal control (ETNOC) for unknown discrete-time (DT) nonlinear systems. First, to achieve parallel control, an augmented nonlinear system (ANS) with an augmented performance index (API) is proposed to introduce the control input into the feedback system. The control stability relationship between the ANS and the original system is analyzed, and it is shown that, by choosing a proper API, optimal control of the ANS with the API can be seen as near-optimal control of the original system with the original performance index (OPI). Second, based on parallel control, a novel event-triggered scheme is proposed, and then a novel ETNOC method is developed using the time-triggered optimal value function of the ANS with the API. The control stability is proved, and an upper bound, which is related to the design parameter, is provided for the actual performance index in advance. Then, to implement the developed ETNOC method for unknown DT nonlinear systems, a novel online learning algorithm is developed without reconstructing unknown systems, and neural network (NN) and adaptive dynamic programming (ADP) techniques are employed in the developed algorithm. The convergence of the signals in the closed-loop system (CLS) is shown using the Lyapunov approach, and the assumption of boundedness of input dynamics is not required. Finally, two simulations justify the theoretical conjectures.

Convergence of a Robust Deep FBSDE Method for Stochastic Control

In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

T-convex T-differential fields and their immediate extensions

Let $T$ be a polynomially bounded o-minimal theory extending the theory of real closed ordered fields. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring and a $T$-derivation. If this derivation is continuous with respect to the valuation topology, then we call $K$ a $T$-convex $T$-differential field. We show that every $T$-convex $T$-differential field has an immediate strict $T$-convex $T$-differential field extension which is spherically complete. In some important cases, the assumption of polynomial boundedness can be relaxed to power boundedness.

ON THE LOWEST EIGENVALUE OF THE FRACTIONAL LAPLACIAN FOR THE INTERSECTION OF TWO DOMAINS

AbstractWe extend a result of Lieb [‘On the lowest eigenvalue of the Laplacian for the intersection of two domains’, Invent. Math.74(3) (1983), 441–448] to the fractional setting. We prove that if A and B are two bounded domains in $\mathbb R^N$ and $\lambda _s(A)$ , $\lambda _s(B)$ are the lowest eigenvalues of $(-\Delta )^s$ , $0<s<1$ , with Dirichlet boundary conditions, there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)$ . Moreover, without the boundedness assumption on A and B, we show that for any $\varepsilon>0$ , there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)+\varepsilon .$

Stochastic MPC With Dynamic Feedback Gain Selection and Discounted Probabilistic Constraints

This paper considers linear discrete-time systems with additive disturbances, and designs a Model Predictive Control (MPC) law incorporating a dynamic feedback gain to minimise a quadratic cost function subject to a single chance constraint. The feedback gain is selected online and we provide two selection methods based on minimising upper bounds on predicted costs. The chance constraint is defined as a discounted sum of violation probabilities on an infinite horizon. By penalising violation probabilities close to the initial time and assigning violation probabilities in the far future with vanishingly small weights, this form of constraints allows for an MPC law with guarantees of recursive feasibility without a boundedness assumption on the disturbance. A computationally convenient MPC optimisation problem is formulated using Chebyshev's inequality and we introduce an online constraint-tightening technique to ensure recursive feasibility. The closed loop system is guaranteed to satisfy the chance constraint and a quadratic stability condition. With dynamic feedback gain selection, the closed loop cost is reduced and conservativeness of Chebyshev's inequality is mitigated. Also, a larger feasible set of initial conditions can be obtained. Numerical simulations are given to show these results.

Triviality of bounded solutions and gradient estimates for nonlinear f-heat equations on complete smooth metric measure spaces

Some qualitative properties of positive smooth solutions to a generalized nonlinear parabolic equation involving f-Laplacian (Lf)(Lf−q(x,t)−∂∂t)w(x,t)=G(w(x,t)), are discussed on Mf×(−∞,+∞), where Mf is a complete smooth metric measure space (with or without boundary), the potential function q(x,t) is smooth at least C1 in x and C0 in t, and G(w(x,t)) is a nonlinear smooth sourcing term. Local and global type space only (elliptic type) gradient estimates are established for this equation under the condition that Bakry-Émery Ricci curvature tensor is bounded from below. As an exploitation of the gradient estimates so derived we obtain a parabolic Harnack inequality and some Liouville type theorems for bounded ancient and eternal solutions. The approach adopted in this paper provides a unified treatment of a large class of nonlinear source terms. To demonstrate this further, cases where G(w)=awρ, a∈R, ρ∈(−∞,0]∪[1,+∞), G(w)=aw|log⁡w|γ,a∈R, a≠0, γ>1 and G(w)=aw(log⁡w)γa≠0, γ≥1 are considered as specific examples. We further show that all the obtained results also hold on weighted manifolds with compact boundary under some lower boundedness assumptions on mean curvature of the boundary and Bakry-Émery Ricci curvature tensor.

Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation

Gradient descent (GD) type optimization schemes are the standard methods to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Such schemes can be considered as discretizations of the corresponding gradient flows (GFs). In this work we analyze GF processes in the training of ANNs with ReLU activation and three layers. In particular, in this article we prove in the case where the distribution of the input data is absolutely continuous with respect to the Lebesgue measure that the risk of every bounded GF trajectory converges to the risk of a critical point. In addition, we show in the case of a 1-dimensional affine target function and a uniform input distribution that the risk of every bounded GF trajectory converges to zero if the initial risk is sufficiently small. Finally, we show that the boundedness assumption can be removed if the hidden layer consists of only one neuron.

Synchronization of Epidemic Systems with Neumann Boundary Value under Delayed Impulse

This paper reports the construction of synchronization criteria for the delayed impulsive epidemic models with reaction–diffusion under the Neumann boundary value. Different from the previous literature, the reaction–diffusion epidemic model with a delayed impulse brings mathematical difficulties to this paper. In fact, due to the existence of second-order partial derivatives in the reaction–diffusion model with a delayed impulse, the methods of first-order ordinary differential equations from the previous literature cannot be effectively applied in this paper. However, with the help of the variational method and an appropriate boundedness assumption, a new synchronization criterion is derived, and its effectiveness is illustrated by numerical examples.

The level sets of functions with bounded criticalsets and bounded Hess+ complements

"We denote by ${\rm Hess}^+(f)$ the set of all points $p\in\mathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ is positive definite. In this paper we prove several properties of real-valued functions of several variables by showing the connectedness of their level sets for sufficiently high levels, under the boundedness assumption on the critical set. In the case of three variables we also prove the convexity of the levels surfaces for sufficiently high levels, under the additional boundedness assumption on the ${\rm Hess}^+$ complement. The selection of the {\em a priori} convex levels, among the connected regular ones, is done through the positivity of the Gauss curvature function which ensure an ovaloidal shape of the levels to be selected. The ovaloidal shape of a level set makes a diffeomorphism out of the associated Gauss map. This outcome Gauss map diffeomorphism is then extended to a smooth homeomorphism which is used afterwards to construct one-parameter families of smooth homeomorphisms of Loewner chain flavor. "

Incremental nonlinear stability analysis of stochastic systems perturbed by Lévy noise

AbstractWe present a theoretical framework for characterizing incremental stability of nonlinear stochastic systems perturbed by either compound Poisson shot noise or finite‐measure Lévy noise. For each noise type, we compare trajectories of the perturbed system with distinct noise sample paths against trajectories of the nominal, unperturbed system. We show that for a finite number of jumps arising from the noise process, the mean‐squared error between the trajectories exponentially converge toward a bounded error ball across a finite interval of time under practical boundedness assumptions. The convergence rate for shot noise systems is the same as the exponentially stable nominal system, but with a tradeoff between the parameters of the shot noise process and the size of the error ball. The convergence rate and the error ball for the Lévy noise system are shown to be nearly direct sums of the respective quantities for the shot and white noise systems separately, a result which is analogous to the Lévy–Khintchine theorem. We demonstrate both empirical and analytical computation of the error ball using several numerical examples, and illustrate how varying the parameters of the system affect the tightness of the bound.

Global Cooperative Output Regulation of Linear Multiagent Systems With Limited Bandwidth

This article investigates the global cooperative output regulation problem of continuous-time linear multiagent systems (MASs), under the communication constraint of limited bandwidth. With a novel scaling function strategy, a distributed protocol based on the sampled and quantized data is designed to solve this problem. Compared with the existing works on quantized control of MASs, not only the boundedness assumption on initial conditions is removed, but also the communication graph is not restricted to be balanced or strongly connected. Moreover, the proposed quantized cooperative output regulation protocol does not rely on any global information of the communication graph. A simulation example is finally presented to illustrate the effectiveness of the proposed quantized cooperative output regulation protocol.

Zonotopic unknown input state estimator for discrete-time linear systems

In this work state estimation for unknown input discrete-time linear systems is tackled in a bounded error context. First, under the boundedness assumption on both process and measurement noise as well as on system’s initial state, the estimation problem is reformulated as a reachability problem. Then, by preprocessing the set-valued expression of the general solution of the state equation and applying zonotopic set computation rules, a non-conservative and computationally tractable algorithm is proposed to characterize efficiently the solution set of this problem. Numerical examples are presented to show the performance of the proposed zonotopic unknown input state estimator.

Adaptive Barrier Sliding-Mode Control Considering State-Dependent Uncertainty

Aiming at the robust control problem of disturbed systems that contain uncertain parameters, a novel control scheme termed adaptive barrier sliding-mode control (ABSMC) is presented. The proposed ABSMC contains an adaptive switching gain based on the barrier function along with a state-dependent assumption of the lumped uncertainty. Distinguished from existing similar works, the obvious advantage of the proposed scheme is that ABSMC enables convergence of the system trajectory and holds it within a predefined vicinity of the sliding surface independent of the lumped uncertainty, without control gain overestimation. Moreover, the conventionally adopted priori constant boundedness assumption of the lumped uncertainty is replaced by a more reasonable state-dependent assumption. The stability analysis of the controlled system is carried out through Lyapunov theorem. The performance improvement of the proposed scheme is validated through comparative simulation.

Dunkl Translations, Dunkl-Type BMO Space, and Riesz Transforms for the Dunkl Transform on $$L^\infty$$

In this paper, we will give some results on the support of Dunkl translations on compactly supported functions. Then we will define Dunkl-type $BMO$ space and Riesz transforms for Dunkl transform on $L^\infty$, and prove the boundedness of Riesz transforms from $L^\infty$ to Dunkl-type $BMO$ space under the uniform boundedness assumption of Dunkl translations. The proof and the definition in Dunkl setting will be harder than in the classical case for the lack of some similar properties of Dunkl translations to that of classical translations. We will also extend the preciseness of the description of support of Dunkl translations on characteristic functions by Gallardo and Rejeb to that on all nonnegative radial functions in $L^2(m_k)$.

A q-Gradient Descent Algorithm with Quasi-Fejér Convergence for Unconstrained Optimization Problems

We present an algorithm for solving unconstrained optimization problems based on the q-gradient vector. The main idea used in the algorithm construction is the approximation of the classical gradient by a q-gradient vector. For a convex objective function, the quasi-Fejér convergence of the algorithm is proved. The proposed method does not require the boundedness assumption on any level set. Further, numerical experiments are reported to show the performance of the proposed method.

Model predictive tracking control of uncertain moving structures with nonaffine saturated actuation: Rotating control force

This article presents a novel scheme for model-based control of multi-input–multi-output (MIMO) systems considering input nonaffinity, nonlinear dynamical effects, and bounded modeling uncertainty. Commonly, model-based control of a nonaffine system is conducted based on an equivalent pseudo-affine expression. However, validity of this approximation necessitates boundedness of inputs and states. Furthermore, existence of truncation errors is inevitable when obtaining the pseudo-affine model. Therefore, robustness to bounded uncertainty should be ensured even in cases where the original system is deterministic. To address the expressed issues, in this study the boundedness assumptions are incorporated in a constrained robust model predictive control (MPC) algorithm. The corresponding MPC scheme is based on construction of cost according to predicted sliding functions over a finite prediction horizon. It should be noted that for the considered class of uncertain nonlinear nonaffine system, obtaining the perquisite robust stability and feasibility conditions is non-trivial. To attain the aforementioned qualities, it is proposed that sliding functions can be expressed as the sum of input-based terms (using pseudo-affine approximation of input gains based on Taylor expansion) and dynamical variation terms. Subsequently, robust stability is ensured by constructing a Lyapunov-based terminal cost. Constraints satisfaction conditions are determined based on calculation of the corresponding feasible region. Numerical simulations for a nonlinear nonaffine mechanical system indicate efficiency of the presented control model. Comparisons with other applicable schemes highlight significant features of the presented algorithm and the attained improvements regarding stability and control feasibility are discussed.

Parameter‐dependent observer‐based feedback compensator design of a space‐time‐varying PDE with application to a class of steelmaking processes

AbstractThis article deals with the problem of observer‐based feedback compensator design for a linear reaction–advection–diffusion equation subject to a time‐varying diffusion coefficient, a time‐varying advection coefficient, and a space‐time‐varying reaction coefficient. To solve such a problem, these coefficients are written in parametric forms under their boundedness assumptions. By considering these parametric forms of the coefficients and the distribution functions for control actuation and non‐collocated measurement, a parameter‐dependent observer‐based feedback compensator is constructed such that the resulting closed‐loop coupled equation exponentially converges to a bounded set of the equilibrium profile in the spatial norm. With the aid of the Lyapunov technique and variants of Poincaré–Wirtinger's inequality, a sufficient condition for the existence of such feedback compensator is presented in the form of convex constraints. Finally, extensive simulation results for a numerical example and a class of steelmaking processes are presented to support the proposed design method. For the collocated measurement case, both theoretical and simulation results show that the proposed observer‐based feedback compensator can provide a better control performance than the static feedback compensator in the presence of measurement disturbances.

Learning Bounded Subsets of Lₚ

We study learning problems in which the underlying class is a bounded subset of L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> and the target Y belongs to L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> . Previously, minimax sample complexity estimates were known under such boundedness assumptions only when p=∞. We present a sharp sample complexity estimate that holds for any p > 4; it is based on a learning procedure that is suited for heavy-tailed problems.