In the present paper we propose and study a simple model of collagen remodeling occurring in latter stage of tendon healing process. The model is an integro-differential equation describing the possibility of an alignment of collagen fibers in a finite time. We show that the solutions may either exist globally in time or blow-up in a finite time depending on initial data. The latter behavior can be related to the healing of injury without the scar formation in a finite time: a full alignment of collagen fibers. We believe that the present model is an essential ingredient of the full description of collagen remodeling.

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A system of a parabolic partial differential equation coupled with ordinary differential inclusions that arises from a closed-loop control problem for a thermodynamic process governed by the Allen-Cahn diffusion reaction model is studied. A feedback law for the closed-loop control is proposed and implemented in the case of a finite number of control devices located inside the process domain basing on the process dynamics observed at a finite number of measurement points. The existence of solutions to the discussed system of differential equations is proved with the use of a generalization of the Kakutani fixed point theorem.

A closed-loop control of a reaction-diffusion type process is introduced. The control system consist of a finite number of control and measurement devices. The measurement devices collect information about the current state of the process. The control devices influence the process, responding to data obtained from the measurement devices. Each control device takes into account the data from all measurement devices. The rule of accounting the data from measurement devices by a single control device involves defining suitable weights for each pair of one control device and one measurement device. A weight reflects how important is a given measurement device to a given control device. The aim of this control system is to bring the process possibly close to a user defined reference state or trajectory.We are interested in a situation where the user can adjust the control system by choice of the control and measurement devices and the weights. For this reason, one of the aims of the preset work is to study the behavior under perturbations of these elements for the mathematical model realizing the above control concept. Moreover, we formulate and justify results concerning existence and uniqueness of solutions of the investigated model. Finally, numerical prototypes illustrating properties of the model are presented.