Two major obstacles against successful chemotherapy of cancer are the cell-cycle-phase dependence of treatment, and the emergence of resistance of cancer cells to cytotoxic agents. One way to understand and overcome these two problems is to apply optimal control theory to mathematical models of cell cycle dynamics. These models should include division of the cell cycle into subphases and/or the mechanisms of drug resistance. We review our relevant results in mathematical modelling and control of the cell cycle and of the mechanisms of gene amplification (related to drug resistance), and estimation of parameters of the constructed models.

The paper deals with the -norm of finite dimensional linear continuous-time periodic (FDLCP) systems, represented in state space. On basis of general formulae for the -norm significantly simpler formulae are derived under the assumption of stability of the system. The proposed method needs to compute the transition matrix on the interval , where T is the period of the system, and it operates only with matrices of finite dimension. An example is given, where the -norm is determined exactly, and the proposed method turns out to be superior to other known methods.

In the paper a controllability problem for a class of time invariant, discrete linear SISO systems with uncertain parameters is discussed. The system under consideration is described by a finite dimensional state - space equation with an interval diagonal state matrix, an interval control matrix, the known output matrix and the two - dimensional uncertain parameters space. From the considerations shown in the paper it turns out, that the general controllability conditions for the discussed discrete system are the same as for the analogical continuous system with uncertain parameters. As an example the controllability problem for an interval parabolic system is discussed, numerical examples are also presented.

In a standard multi-input, multi-output linear time invariant (MIMO LTI) continuous-time system S(A,B,C) the classical notion of the Smith zeros does not characterize fully the output-zeroing problem nor the zero dynamics. The question how this notion can be extended and related to the state-space methods is discussed. Nothing is assumed about the relationship of the number of inputs to the number of outputs nor about the normal rank of the underlying system matrix. The proposed extension treats multivariable zeros (called further the invariant zeros) as the triples (complex number, nonzero state-zero direction, input-zero direction). Such a treatment is strictly connected with the output zeroing problem and in that spirit the zeros can be easily interpreted even in the degenerate case (i.e., when any complex number is such a zero).
A simple sufficient and necessary condition of nondegeneracy is presented. The condition decomposes the class of all systems S(A,B,C) such that B0 and C0 into two disjoint subclasses: of nondegenerate and degenerate systems. In nondegenerate systems the Smith zeros and the invariant zeros are exactly the same objects which are determined as the roots of the so-called zero polynomial. The degree of this polynomial equals the dimension of the maximal (A,B)-invariant subspace contained in ker C, while the zero dynamics are independent upon control vector. In degenerate systems the zero polynomial determines merely the Smith zeros, while the set of the invariant zeros equals the whole complex plane. The dimension of the maximal (A,B)-invariant subspace contained in ker C is strictly larger than the degree of the zero polynomial, whereas the zero dynamics essentially depend upon control vector.

In this paper a new sliding mode control algorithm for the third order uncertain, nonlinear and time-varying dynamic system subject to unknown disturbance and input constraint is proposed. The algorithm employs a time-varying switching plane. At the initial time , the plane passes through the point determined by the system initial conditions in the error state space. Afterwards, the plane moves with a constant velocity to the origin of the space. The plane is selected in such a way that the integral of the absolute value of the system error over the whole period of the control action is minimized and the input constraint is satisfied. By this means, the reaching phase is eliminated, insensitivity of the system to the external disturbance is guaranteed from the very beginning of the proposed control action and fast, monotonic error convergence to zero is achieved.