Alexander Y. Khapalov

Controllability Properties of a Vibrating String with Variable Axial Load Only

We show that the set of equilibrium-like states (yd,0) of a vibrating string which can approximately be reached in the energy space H10 (0,1)× L2(0,1) from almost any non-zero initial datum, namely, (y0,y1) e (H2(0,1)∩H10(0,1))× H1(0,1), (y0,y1) ≠ (0,0) by varying its axial load only is dense in the subspace H10 (0,1)× {0} of this space. This result is based on a constructive argument and makes use of piecewise constant-in-time control functions (loads) only, which enter the model equation as coefficients.

Mobile point controls versus the locally distributed ones for the controllability of the semilinear parabolic equation

It is well known that a rather general semilinear parabolic equation with globally Lipschitz nonlinear term is both approximately and exactly null-controllable in L/sup 2/ (/spl Omega/), when governed in a bounded domain by locally distributed controls. We show that, in fact, in one space dimension the very same results can be achieved by employing at most two mobile point controls with support on the curves properly selected within an arbitrary subdomain of Q/sub T/ = (0,1) /spl times/ (0,T). We show that such curves can be described by certain differential inequalities and provide explicit examples.

Well-posedness of a 2D swimming model governed in the nonstationary Stokes fluid by multiplicative controls

We introduce and discuss the well-posedness of a schematic simplified mathematical model of an abstract object ‘swimming’ in the 2D nonstationary Stokes fluid. The object consists of finitely many subsequently connected small sets, each of which can act upon a pair of the adjacent sets in a rotation fashion with the purpose to generate its fish-like or rowing motion. The structural integrity of the object is maintained by respective elastic Hookes forces. Such models are of interest in biological and engineering applications dealing with propulsion systems in fluids. Mathematics-wise, the model equations are described by a complex highly nonlinear hybrid system of partial differential equations and ordinary differential equations, which include a fluid equation and an equation describing the motion of the object in it. Our study is linked to viewing the swimming process as the one governed by multiplicative (bilinear) controls.

Addendum to The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation

We introduce and investigate the well-posedness of a model describing the self-propelled motion of a small abstract swimmer in the 3-D incompressible fluid governed by the nonstationary Stokes equation, typically associated with low Reynolds numbers. It is assumed that the swimmer’s body consists of finitely many subsequently connected parts, identified with the fluid they occupy, linked by rotational and elastic Hooke forces. Models like this are of interest in biological and engineering applications dealing with the study and design of propulsion systems in fluids.

Multiplicative controllability for semilinear reaction–diffusion equations with finitely many changes of sign

We study the global approximate controllability properties of a one dimensional semilinear reaction-diffusion equation governed via the coefficient of the reaction term. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Our goal is to show that any target state

Geometric Aspects of Controllability for a Swimming Phenomenon

u^*\in H_0^1 (0,1)

The Well-Posedness of a 2-D Swimming Model

, with as many changes of sign in the same order as the given initial data

Controllability of the Wave Equation Governed by Mobile Point Controls

u_0\in H^1_0(0,1)

Degenerate Sensors in Source Localization and Sensor Placement Problems

, can be approximately reached in the

Controllability of the Semilinear Heat Equation with a Sublinear Term and a Degenerate Actuator

L^2 (0,1)