# Bronisław JakubczykInstitute of Mathematics Polish Academy of Sciences

Bronisław Jakubczyk

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78

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Introduction

**Skills and Expertise**

## Publications

Publications (78)

Let $M$ be a free module of rank $m$ over a commutative unital ring $R$ and let $N$ be its free submodule. We consider the problem of when a given element of the exterior product $\Lambda^pM$ is divisible, in a sense, over elements of the exterior product $\Lambda^r N$, where $r\le p$. Precisely, we give conditions under which a given $\eta\in\Lamb...

Let $E$ be a vector bundle over a suitable differential manifold $M$ and let $\wedge^p E$ denote $p$-exterior product of $E$. Given sections $\omega_1,\dots,\omega_k$ of $E$ and a section $\eta$ of $\wedge^p E$, we consider the problem if $\eta$ can be written in the form $$\eta=\sum \omega_i\wedge\gamma_i,$$ where $\gamma_i$ are sections of $\wedg...

We outline a method of finding sinusoidal controls which assure a semi-exact path following. The method is designed primarily for non-admissible paths and controllable nonholonomic systems. The fastly oscillating sinusoidal controls generate (in approximate sense) new control variables which enable to control new directions. In this way the system...

We study dynamic pairs (X,ν) where X is a vector field on a smooth manifold M and ν ⊂ TM is a vector distribution, both satisfying certain regularity conditions. We construct basic invariants of such objects and solve the equivalence problem. In particular, we assign to (X, ν) a canonical connection and a canonical frame on a certain frame bundle....

We study local controllability and optimal control problems for invertible
discrete-time control systems. We present second order necessary conditions for
optimality and sufficient conditions for local controllability. The conditions
are stated in geometric terms, using vector fields naturally associated to the
system. The Hessian of the optimal pr...

We consider the motion planning problem, where the nonholonomic constraints are given by a strong-bracket-generating distribution. Approximating a nonadmissible trajectory by an admissible one in the sub-Riemannian setting, we prove a theorem which provides an exact asymptotic estimate of the “interpolation entropy” in the case of a free nilpotent...

We study generic distributions D⊂TM of corank 2 on manifolds M of dimension n⩾5. We describe singular curves of such distributions, also called abnormal curves. For n even the singular directions (tangent to singular curves) are discrete lines in D(x), while for n odd they form a Veronese curve in a projectivized subspace of D(x), at generic x∈M. W...

We introduce invariants of control-affine systems which we call curvatures. They are defined by the drift and the control distribution, given by the system. The curvatures allow us to analyse the variational equation along a given trajectory, as well as existence of conjugate points.

We study a separability problem suggested by mathematical description of bipartite quantum systems. We consider hermitian 2-forms on the tensor product H = K ⊗ L, where K, L are finite-dimensional complex spaces. Such a form is called separable if it is a convex combination of hermitian tensor products σ*p⊙σp of 1-forms σp on H that are product for...

We show that the Hamiltonian formalism, as used for optimal control problems, is a natural tool for studying the feedback
equivalence problem and for constructing invariants. The most important invariants (covariants) are the critical curves or
trajectories of the system, also called extremals or singular curves. They are obtained as the curves sat...

Given a control system with an output, we give explicit conditions on the system which guarantee that it admits a Hamiltonian structure. The conditions are stated directly in terms of vector fields and observation functions defined by the system.
en a control system with an output, we give explicit conditions on the system which guarantee that it a...

We define bifurcations of control-affine systems in the plane and classify all generic 1-parameter bifurcations at control-regular points. More precisely, we classify topological bifurcations of invariants of usual feedback equivalence. Such bifurcations form six different classes: two bifurcations of equilibrium sets, two bifurcations of critical...

Symmetric functions of critical Hamiltonians, called symbols, were used in such problems of nonlinear control as the characterization of symmetries and feedback invariants. We derive here a stabilizability condition in the class of almost continuous feedback controls based on symbols. The methodology proposed consists of defining a selector of the...

We study nonlinear control systems in the plane, affine with respect to control. We introduce two sets of feedback equivariants forming a phase portrait PP and a parameterized phase portrait PPP of the system. The phase portrait PP consists of an equilibrium set E, a critical set C (parameterized, for PPP), an op- timality index, a canonical foliat...

This paper follows a recent classification, obtained by the authors, of generic bifurcations of 1-parameter families of planar control systems. In this paper, we analyze all bifurcations of planar systems with quadratic nonlinearities, in particular those not appearing among generic ones. To illustrate our results, we discuss briefly bifurcations o...

We define bifurcations of control-affine systems in the plane and classify all generic 1-parameter bifurcations with nonvanishing control field. More precisely, we classify topological bifurcations of invariants of usual feedback equivalence. There are six such bifurcations: two bifurcations of equilibrium sets, two bifurcations of critical sets an...

We consider the complex 0 → Λ0(M; E) ∂ ω→ Λ1(M; E) ∂ ω→ ⋯ ∂ω→ Λ m(M; E), where E is a finite-dimensional vector bundle over a suitable differential manifold M, Λq(M; E) denotes the space of all smooth or real analytic or holomorphic sections of the q-exterior product of E and ∂ω(η) := ω Λ η for ω ∈ λ1(M; E). We give sufficient and necessary conditi...

We prove that any 1-parameter family of corank 1 distributions (or Pfaff equations) on a compact manifold Mn is trivializable, i.e., transformable to a constant family by a family of diffeomorphisms, if all distributions of the family have the same characteristic line field. The characteristic line field is a field of tangent lines which is invaria...

We prove that any 1-parameter family of corank 1 distributions (or Pfaff equations) on a compact manifold M n M^{n} is trivializable, i.e., transformable to a constant family by a family of diffeomorphisms, if all distributions of the family have the same characteristic line field. The characteristic line field is a field of tangent lines which is...

A differential 1-form on a (2k+1)-dimensional manifold M defines a singular contact structure if the set S of points where the contact condition is not satisfied, S={p∈M:(ω∧(dω) k (p)=0}, is nowhere dense in M. Then S is a hypersurface with singularities and the restriction of ω to S can be defined. Our first theorem states that in the holomorphic,...

Let ω be a local nonvanishing differential 1-form on a (2k+1)-dimensional manifold with structurally smooth hypersurface S of singular points (the points at which us A (dw)k vanishes). We prove that in the holomorphic, real-analytic, and C∞ categories, the Pfaffian equation (ω) is determined, up to a diffeomorphism, by its restriction to S, a canon...

We consider dynamic transformations of systems which are defined
as transformations depending on the variables and their time derivatives
up to finite order. Two systems are called dynamically equivalent if
they have the same sets of trajectories, up to dynamic transformations.
A system is called free if it is dynamically equivalent to a system
whi...

In a space of nonlinear ordinary differential equations we consider two equivalence relations. Static equivalence means equivalence up to nonlinear transformations of dependent variables. A much weaker dynamic equivalence uses transformations which depend on dependent variables and their derivatives up to finite order (this relation is closely rela...

In this note we consider a problem of equivalence of nonlinear systems described by a finite set of real variables satisfying a system of ordinary differential equations of finite order. Two systems are called equivalent (resp. weakly equivalent) if there exist nonlinear transformations so that the variables of one system are functions of the varia...

In this note we consider a problem of equivalence of nonlinear systems described by a finite set of real variables satisfying a system of ordinary differential equations of finite order. Two systems are called equivalent (resp. weakly equivalent) if there exist nonlinear transformations so that the variables of one system are functions of the varia...

This paper presents a geometric study of controllability for discrete-time nonlinear systems. Various accessibility properties are characterized in terms of Lie algebras of vector fields. Some of the results obtained are parallel to analogous ones in continuoustime, but in many respects the theory is substantially different and many new phenomena a...

We consider local feedback equivalence of control-affine systems in the plane. We present a general normal form for such systems with scalar control. In this form a polynomial of one variable appears, with coefficients which are functions of another variable. We show that these coefficients (called functional parameters) are invariants. This gives...

In a recent paper we showed that if, in a neural network given by a discrete Hopneld model (nonsymmetric), the product of weights in any cycle of neurons is nonnegative, then each trajectory converges to a stable state with probability one. This result is generalized here to a much larger class of systems which can model interactions in nonsymmetri...

We consider local feedback equivalence and local weak feedback equivalence of control systems. The later equivalence is up to local coordinate changes in the state space, local feedback transformations, and state dependent changes of the time scale. We show that, under such equivalence, there are only five nonequivalent local canonical forms (some...

This paper presents a geometric study of controllability for discrete-time nonlinear systems. Various accessibility properties are characterized in terms of Lie algebras of vector fields. Some of the results obtained are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena...

We investigate the effect of sampling on linearization for continuous time systems. It is shown that the discretized system is linearizable by state coordinate change for an open set of sampling times if and only if the continuous time system is linearizable by state coordinate change. Also, it is shown that linearizability via digital feedback imp...

We define and study a notion of a singular point for observed dynamical systems with no controls. Lists of normal forms for
such singularities are given in both general and generic cases. One class of normal forms corresponds to singularities which
appear in catastrophe theory.

For a nonsymmetric threshold network equipped with an asynchronous dynamics, we show that if the product of weights in any cycle of units is nonnegative, then each trajectory converges to a stable state with probability one. We also show that such networks have a natural feed- forward layer structure and stability is achieved in a hierarchical orde...

We study the effect of sampling on the standard technique of controlling a system by reduction to a linear form. It is shown that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback.

We give necessary and sufficient conditions for a nonlinear discrete-time system to be locally linearizable by a change of state coordinates and a feedback. The conditions are expressed in terms of controllability distributions associated to the system.

We give necessary and sufficient conditions for a causal operator to have a local realization of class C**k(k equals 1, . . . , infinity , omega ) of the form dx/dt equals f(x, u), y equals h(x). We show that two minimal local realizations of the same response map are locally diffeomorphic.

The aim of this paper is to give a short presentation of three approaches that exist in the nonlinear realization theory of continuous-time systems. They differ from each other in the way of describing the input-output behaviour of the system. The first describes this behaviour by a general causal operator mapping input signals to output signals. W...

Necessary and sufficient conditions for the existence of either analytic or smooth symmetric realizations of nonlinear input-output maps are given. It is also shown that two minimal realizations are diffeomorphic.

A linear control system with delays of neutral type is characterized by a pair of matrices over the ring R0(s) of proper rational functions which can be embedded into the field IR(s) of rational functions. This leads to a natural concept of IRn (s)-controllability and to canonical forms related to those known for matrices over reals as Luenberger a...

A linear control system with time-delay of neutral type is considered. Under the assumption that the state variables are accessible for feedback the stabilization problem is examined. The general type of feedback considered consists of a proportional term plus a dynamic term multiplied by the delay element. Sufficient conditions for the existence o...

A local differential 1-form ω on a 3-manifold M defines a local singular contact structure if the set S of noncontact points, S={p∈M:(ω∧(dω) k )(p)=0}, is nowhere dense in M. Then S is a hypersurface with singularities and the restriction of ω to S can be defined. Our results state that if S is regular then: (i) in the holomorphic category the loca...