Novel approach to investigation and control of nonlinear nonstationary processes: application to environments and biomedical engineering

Further development of structural interactive control concepts for nonstationary processes should involve nonlinear gradient diffusion and nonlinear sources. The corresponding nonstationary equations are considered on a set of relevant self-similar solutions, depending on the physical, biological, chemical, and other parameters. We use conservation laws to reduce obtained problems to interactive structural and parametric control of the phase-plane transformations for associated nonlinear ordinary differential equations. Such control of critical points and critical levels sets of the corresponding finite-dimensional function (Hamiltonian) makes the control of transition states' (transients') structure possible, determining the process evolution. This approach allows predicting new nonlinear effects: localization or periodicity of transition waves, suppression of chaotization in transition processes, etc. We consider some applications of the nonlinear spectral theory to these control problems. The approaches mentioned above can be successfully used in environmental sciences and bioengineering.