In this paper we present the extraproximal method for computing the Stackelberg/Nash equilibria in a class of ergodic controlled finite Markov chains games. We exemplify the original game formulation in terms of coupled nonlinear programming problems implementing the Lagrange principle. In addition, Tikhonov’s regularization method is employed to ensure the convergence of the cost-functions to a Stackelberg/Nash equilibrium point. Then, we transform the problem into a system of equations in the proximal format. We present a two-step iterated procedure for solving the extraproximal method: (a) the first step (the extra-proximal step) consists of a “prediction” which calculates the preliminary position approximation to the equilibrium point, and (b) the second step is designed to find a “basic adjustment” of the previous prediction. The procedure is called the “extraproximal method” because of the use of an extrapolation. Each equation in this system is an optimization problem for which the necessary and efficient condition for a minimum is solved using a quadratic programming method. This solution approach provides a drastically quicker rate of convergence to the equilibrium point. We present the analysis of the convergence as well the rate of convergence of the method, which is one of the main results of this paper. Additionally, the extraproximal method is developed in terms of Markov chains for Stackelberg games. Our goal is to analyze completely a three-player Stackelberg game consisting of a leader and two followers. We provide all the details needed to implement the extraproximal method in an efficient and numerically stable way. For instance, a numerical technique is presented for computing the first step parameter (λ) of the extraproximal method. The usefulness of the approach is successfully demonstrated by a numerical example related to a pricing oligopoly model for airlines companies

The problem of observability for systems with unknown inputs is revised. The sufficient and necessary conditions are used for the design of an observer for linear systems with bounded unknown inputs. To realize the observation of the state, a second-order sliding-mode observer is suggested to be applied. Such an observer provides a robust estimate of the state vector in a finite time, without filtration. The design is based on the concept of the hierarchical output injection maintaining zero value for output tracking error at each level of the hierarchy. The equivalent control is used to identify the unknown inputs. A numerical example illustrates the effectiveness of the suggested technique.

The behaviour of a standard super-twist controller under stochastic perturbations, when its dynamic is governed by the stochastic differential equation of Itô type with discontinuos right-hand side, is studied. The suggested analysis is based on the Lyapunov functions $V_{t}=V(x_{t},y_{t})$ (Polyakov-Poznyak, Moreno-Osorio, Orlov and Utkin) designed for the stability analysis of the deterministic version of super-twist controllers. The major finding is that under stochastic (in fact, unbounded) perturbations the special selection of a gain-parameter of such controller, making it depending on $V_{t}$ and its gradient $\frac{\partial V}{\partial _{y}}(x_{t},y_{t})$, provides the controller with an adaptivity property and guarantees the mean square convergence of V(t) into the prespecified zone around the origin which depends on the diffusion parameter of stochastic noise, the upper estimate of the second derivative $\frac{\partial^{2}V}{\partial_{y}{2}}$ as well as on the parameters of the controller.