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The problem of design and analysis of synchronization control circuits is a challenging task for many applications: satellite navigation, digital communication, wireless networks, and others. In this article the Charge-Pump Phase-Locked Loop (CP-PLL) electronic circuit, which is used for frequency synthesis and clock generation in computer architectures, is studied. Analysis of CP-PLL is not trivial: full mathematical model, rigorous definitions, and analysis still remain open issues in many respects. This article is devoted to development of a mathematical model, taking into account engineering aspects of the circuit, interpretation of core engineering problems, definition in relation to mathematical model, and rigorous analysis.

The paper addresses stabilization under communication errors and limited data rate by means of controllers with bounded (as time progresses) computational powers. Discrete-time partially observed linear systems perturbed by stochastic additive disturbances are studied. The sensor signals are communicated to the controller over a noisy finite capacity link modeled as a stochastic discrete memoryless channel (DMC). We study stabilization in probability. It is shown that stability cannot typically be achieved by means of a finite memory decoder-controller so far as the boundary of the corresponding stabilizability domain is given by the zero-error capacity of the channel, which is typically zero. At the same time, stability can be achieved with keeping the expected values of the consumed computational resources bounded: the boundary of the relevant stabilizability domain is given by the ordinary Shannon capacity of the noisy channel, which is typically nonzero.

We consider the optimal control problem for the non-linear Goursat-Darboux equations in the case when the quality functional depends only on the phase variable. We show that a natural analogue of the 'traditional' criterion for the existence of an optimal control for this problem is not valid. Under certain assumptions on the equations we obtain necessary and sufficient conditions under which any functional of the type mentioned above attains a minimum. The traditional condition (the convexity of the set of contingencies) is supplemented here by the following essential condition: the right-hand side of the differential equation is affine with respect to each of the two lowest derivatives of the phase variable.

A non-holonomic constant-speed robot travels in an unknown maze-like environment cluttered with complex obstacles. Through the obstacle-free part of the plane, the robot should autonomously arrive at the isoline where an unknown scalar field assumes a given value. Afterwards, it should track the obstacle-free part of the isoline. The robot has access only to the field value at the current location and the distance from this location to the obstacles. We present a hybrid nonlinear navigation law that solves this mission. The law does not use estimation of the field gradient and is non-demanding with respect to both computation and motion. The non-local convergence of the proposed algorithm is rigorously justified and confirmed by computer simulation tests.