This paper investigates the open-loop, closed-loop, and weak closed-loop solvability of discrete-time linear-quadratic (LQ) control problems. First, we address systems that are exclusively open-loop solvable, for which conventional approaches based on the Riccati equation fail to yield a linear state feedback representation of the open-loop control. For such cases, we develop a novel characterization of open-loop solvability employing a perturbation methodology. Second, we establish a weak closed-loop strategy that generates an open-loop optimal control expressible in linear state feedback form. To demonstrate the practical relevance of our theoretical framework, we present two numerical examples that validate the derived results.
With the growing interest in fractional optimal control problems, developing robust numerical methods to solve general multi-dimensional cases efficiently and accurately remains a challenge. This paper proposes a class of hybrid direct methods for solving fractional optimal control problems governed by the Caputo and Atangana–Baleanu–Caputo fractional derivatives. The approach employs a collocation scheme to discretize the state and control variables, while approximating the fractional dynamics using a second-order one-point quadrature rule. Inspired by the principle of control parameterization, we further partition the control values at the collocation points into subsets, each governed by a single decision variable. Compared with standard collocation methods, this strategy reduces the dimensionality of the control input without compromising solution accuracy. Numerical experiments on three benchmark problems demonstrate the superior computational efficiency and robustness of the proposed methods.
We present a semi-analytical solution to the optimal stopping problem of the maximisation of the expected reward representing the difference between the maximum max{ X^1_τ, X^2_τ} and the positive costs c τ over all stopping times τ with finite expectation of two mean-reverting Ornstein–Uhlenbeck (O–U) processes X^i = (X^i_t)_t ≥ 0 , for i = 1, 2 , driven by constantly correlated standard Brownian motions. The method of proof consists of embedding the initial problem into an optimal stopping problem for the two-dimensional diffusion O–U process (X^1 - X^2, X^1 + X^2) . The optimal stopping time is shown to be the first time at which the difference process X^1 - X^2 exits a region formed by continuous stochastic boundaries depending on the current value of the sum process X^1 + X^2 . These boundaries are characterised as a unique solution to the associated system of (fixed-point) nonlinear Fredholm-type integral equations. The result is applied to determine the best constants in the sharp forms of the corresponding maximal inequalities.
This paper investigates pantograph Hilfer fractional stochastic systems with time delays driven by sub-fractional Brownian motion in a finite-dimensional setting. The primary objective is to establish the averaging principle and investigate the Ulam–Hyers stability of the considered system. The existence and uniqueness of solutions are first established by employing the Banach contraction principle and the M ö nch fixed-point theorem. Subsequently, the averaging principle and Ulam–Hyers stability are derived using Jensen’s inequality, the Cauchy–Schwarz inequality, and the Gronwall–Bellman inequality. Finally, using suitable mathematical skills, a numerical example is presented to illustrate the applicability and effectiveness of the proposed theoretical results.
We introduce s-convex b-metric spaces, a framework unifying the nonlinear geometry of b-metrics with s-weighted convexity, addressing a gap where previous convexity notions were limited to linear or m-structures. We establish fixed point theorems for Banach-type, Kannan-type, and Čirić-type contractions under explicit conditions linking the contraction parameters, the b-metric coefficient, and the parameter s. The influence of s on the admissibility of these conditions and on the associated iterative processes is investigated in detail. A theoretically justified bisection-based procedure is developed for selecting feasible values of s, together with a proof of its correctness and numerical validation. A comparative analysis distinguishes our s-convexity from classical, m-convex, and (θ ,δ ) -convex structures. The theoretical results are complemented by applications to nonlinear Fredholm integral equations and boundary value problems, where the effect of the parameter s on the convergence of the corresponding iterations is illustrated numerically. The resulting framework provides a connection between generalized convex structures, fixed point theory, and iterative methods in b-metric spaces.
