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In this paper, we study the approximate controllability of semilinear impulsive functional differential inclusions with nonlocal conditions. Analytic semigroup theory and α-norm arguments are employed to ensure that the obtained results can be applied to the systems involving spatial derivatives. In particular, the compactness condition or Lipschitz condition for the function g in the nonlocal conditions appearing in literature is not required here. An example is provided to illustrate the application of the obtained results.

In this paper, the time-optimal control problem for (2 × 2)-co-operative hyperbolic system with variable coefficients and involving Laplace operator is considered. This problem is, steering the initial state ((y(0), y′(0)) so that the observation z(t) hitting a given target set in minimum time. For different cases of the observation, the time-optimal controls are characterized in terms of the adjoint, this characterization (in particular case of the space of controls) is used to derive specific properties of the optimal control (bang-bangness, uniqueness, etc.). These results are extended to the case of (n × n)-co-operative hyperbolic system.

The purpose of this article is to establish conditions under which a positive quadratic differential form is topologically equivalent to its principal part defined by Newton polyhedra. The problem is to study the simultaneous behavior of two foliations in the plane having a common point as a singularity.

Two classes of robust control problems associated with a class of nonlinear parabolic type systems with multiple time-varying delays are considered, in order to take an account of the influence of noises in data. The time-varying delays are given in the integral form and appear in the state equations. The controls appear either in the forcing terms or in the initial condition. First, the robust control problems are formulated. Afterwards the existence of an optimal solution is proved and necessary conditions of optimality are obtained. Finally the conditions of the uniqueness of the optimal solution are derived.

In this work, we introduce the notion of extended holonomy and discuss some of its properties. As an application of it, we consider vanishing projective holonomy groups of germs of vector fields with zero (n − 1)-jet and prove, under some genericity assumptions, that the corresponding vanishing holonomy groups are topologically conjugated for orbitally topologically equivalent germs of vector fields.

A Kirchhoff-type equation describing the transversal vibrations of a beam is considered. The beam is clamped to a rigid base at one part of its edge and free at the remaining part. On the free part, it is subject to a feedback involving fractional derivatives instead of the classical velocity of the deflection and angular velocity. In presence of an external nonlinear source we prove that solutions blow up at a finite time.

The phase portrait of a control-affine system, being an almost complete feedback invariant, carries the information on time-optimality and controllability structure of the system. In the paper, we deal with generic control affine systems and 1-parameter families of such systems around stationary points of their controllable part. We classify the portraits of such systems up to homeomorphisms of the state space. In the case of 1-parameter families, the classification embraces bifurcations, which can appear when changing the parameter. The figures illustrating these phenomena are included.

We study rational approximations of the transfer function $$\widehat P$$ of a uniform or nonuniform viscoelastic rod undergoing torsional vibrations that are excited and measured at the same end. The approximation is to be carried out in a way that is appropriate, with respect to stability and performance, for the construction of suboptimal rational stabilizing compensators for the rod. The function $$\widehat P$$ can be expressed as $$\widehat P(s) = s^{ - 2} g(\beta ^2 (s))$$ , where g is an infinite product of fractional linear transformations and β is a (generally transcendental) function that characterizes a particular viscoelastic material. First, g(β2) is approximated by its partial products g N(β2). For relevant values of β2, convergence rates for g N are analyzed in detail. Convergence suitable for our problem requires the introduction of a new irrational convergence factor, which must be approximated separately. In addition, the fractional linear factors in β2(s) that appear in g N(β2(s)) must be replaced by something rational. When the damping is weak it is possible to do this by separating the oscillatory modes from the “creep” modes and ignoring the latter; in general, this step remains incomplete. Some numerical data illustrating all the stages of the process as well as the final results for various viscoelastic constitutive relations are presented.

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