Fritz Colonius

Invariance Entropy for Control Systems

For continuous time control systems, this paper introduces invariance entropy as a measure for the amount of information necessary to achieve invariance of weakly invariant compact subsets of the state space. Upper and lower bounds are derived; in particular, finiteness is proven. For linear control systems with compact control range, the invariance entropy is given by the sum of the real parts of the unstable eigenvalues of the uncontrolled system. A characterization via covers and corresponding feedbacks is provided.

Output least squares stability in elliptic systems

In this paper the stability of the solutions of parameter estimation problems in their output least squares formulation is analyzed. The concepts of output least squares stability (OLS stability) is defined and sufficient conditions for this property are proved for abstract elliptic equations. These results are applied to the estimation of the diffusion, convection, and friction coefficient in second-order elliptic equations inℝn,n=2, 3. Results on Tikhonov regularization in a nonlinear setting are also given.

Some aspects of control systems as dynamical systems

AbstractConsider the smooth control system(C)

The Lyapunov spectrum of families of time-varying matrices

\dot x = X_0 (x) + \sum\limits_{i = 1}^m {u_i X_i (x)}

Linear control semigroups acting on projective space

on a manifold M with admissible controlsu εU={u:R →U, locally integrable} and compact control spaceU ⊂Rm. Associated with (C) is a dynamical system whereθt is the shift byt εR to the right onU, andϕ(t, x, u) is the solution of (C) at timet εR with initial condition ϑ(0,x, u) = x, under the control action of uεU We discuss some connections between control properties of (C) and basic notions for dynamical systems, such as topological mixing, chain recurrence, recurrence, invariant (ergodic) measures, and their support. It turns out that these concepts for (D) are related to the control sets and chain control sets of (C): A setD ⊂M is a control set of (C) iff the liftD=cl{(u, x) εU ×M,ϕ(t, x, u) ε D for allt εR} toU×M is a maximal topologically mixing (transitive) component ofφ, similarly for the lifts of chain control sets and the components of the chain recurrent set ofφ. Furthermore, ifμ is an ergodic, invariant measure ofφ, thenπM(suppμ) ⊂=D for some control setD ⊂M, and the pointsx ε M that are contained in control sets, are the projections ontoM ofφ-recurrent points.

Six lectures on dynamical systems

AbstractFor smooth nonlinear systems

Controllability and stabilization of one-dimensional systems near bifurcation points

\dot x(t) = X_0 (x(t)) + \sum\limits_{i = 1}^r {u_i (t)X_i (x(t)),}