JDCS vol. 2 (1996) #4

JDCS, vol. 2, #4 (1996), pp. 449-598

1. Dongho Chae, O.Yu. Imanuvilov, Sang Moon Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions, JDCS vol. 2, No. 4, p. 449--484: This paper studies the problem of exact boundary controllability of second-order semilinear parabolic equations when the control is under Neumann's boundary conditions. For a nonlinear term with a sublinear growth we prove the global null-controllability and for the supperlinear growth case we prove the local exact controllability of the equations.
2. G. Chakvetadze, On the ergodicity of a Model of Drilling, JDCS vol. 2, No. 4, p. 485--502: In this paper a one-dimensional map modeling the process of drilling is studied. The sufficient conditions for the existence of absolutely continuous invariant probability measure and ergodicity are established.
3. A.V. Sarychev, H. Nijmeijer, Extremal Controls for Chained Systems, JDCS vol. 2, No. 4, p. 503--528: To study a time-optimal control problem with bounded controls and an optimal control problem with a quadratic functional for the so-called 2-chained control system $$ \dot{x}_{1}=u_{1},\ \dot{x}_{2}=u_{2},\ \dot{x}_{3}=x_{2}u_{1},\ \ldots,\ \dot{x}_{n}=x_{n-1}u_{1}, $$ we start with the first-order optimality condition, the Pontryagin maximum principle, determine normal, singular and abnormal extremals, provide a complete description of the switching structure for bang-bang extremals of the time-optimal problem, describe their branchings, and derive Jacobi-type second-order optimality conditions for the problem with the quadratic functional. Finally we discuss the relations between 2-chained and power form systems.
4. G. Freiling, G.Jank, Existence and Comparison Theorems for Algebraic Riccati Equations and Riccati Differential and Difference Equations, JDCS vol. 2, No. 4, p. 529--548: We present comparison and global existence theorems forsolutions of generalized matrix Riccati differential and difference equations. Moreover we obtain existence and comparison results for the maximal solutions of the corresponding generalized algebraic Riccati equations. For the symplectic matrix Riccati differential equation we derive sufficient conditions ensuring the global existence of the solutions of the corresponding initial value problems.
5. V.P. Kostov, The minimal number of generators of a matrix algebra, JDCS vol. 2, No. 4, p. 549--556: We give sufficient conditions for a subalgebra (the operations are multiplications and linear combinations over ${\C}$) of $\gl(n,{\C})$ to be generated by two matrices. We show that every algebra can be generated by at most $[n^2/4]+1$ matrices and that all numbers from 1 to $[n^2/4]+1$ can be minimal necessary numbers of generators for some algebra.
6. S. Aranson, E. Zhuzhoma, Maier's theorems and geodesic laminations of surface flows, JDCS vol. 2, No. 4, p. 557--582: We prove that a nontrivial recurrent semitrajectory has an arational asymptotic direction for flows on any hyperbolic surface. To prove the arationality we apply the structure of the stabilizer of a point of the circle at infinity. For a quasiminimal set (a closure of a nontrivial recurrent trajectory) of a flow we construct the corresponding geodesic lamination (the so called ``geodesic framework''). To describe the geodesic framework of quasiminimal sets of flows on the hyperbolic surface of finite genus, we apply Maier's theorem (which states that if one nontrivial recurrent trajectory belongs to the limit set of another nontrivial recurrent trajectory, then the second nontrivial recurrent trajectory belongs to the limit set of the first one). We get the list of types of trajectories of a quasiminimal set that generalizes Cherry's description in the case of the flow on a compact surface. However, we show that the Maier's theorems are not valid for flows on a surface of infinite genus. Geodesic laminations allow us to describe all virtual asymptotic directions of all nontrivial recurrent semitrajectories (formally, we identify the set of directions with some set $\ORA(\Gamma )$ of points of the circle at infinity). We prove that the geodesic framework of the quasiminimal set is determined by its any asymptotic direction. We consider dynamical properties of some subsets of the set $\ORA(\Gamma )$. Orientable geodesic laminations also allow us to classify the set of all virtual asymptotic directions into two classes $\ORA_{1}(\Gamma )$, $\ORA_{2}(\Gamma )$, and we show that different asymptotic directions have different dynamic properties. Namely, we prove that a positive semitrajectory of any arational transitive flow has asymptotic directions from $\ORA_{1}(\Gamma )$ (resp., from $\ORA_{2}(\Gamma )$) if and only if this semitrajectory belongs to the nontrivial recurrent trajectory in both directions (resp., belongs to the $\alpha$-separatrix of some saddle).
7. G. H\"ackl, K.R. Schneider, Controllability near Takens--Bogdanov points, JDCS vol. 2, No. 4, p. 583--598: We study controllability properties of the control-affine system whose uncontrolled part coincides with a universal unfolding of a Takens--Bogdanov singularity. The main result is that the qualitative behavior of the control system can be different from the behavior of all systems with constant control functions in the following sense. There are parameter regions and control ranges such that for constant controls there is no homoclinic orbit, while there exists a ``controlled homoclinic'' orbit corresponding to a nonconstant control function.