Colloquium

Abstract: In the first part of this talk we will recall the fundamental notion of the generalized tangency of a vector v to a set S at x in S in the sense of Bouligand(1930). The set of all such vectors v is said to be the tangent cone T_xS to S at x.
This notion has a strong unifying effect in the theory of a flow invariant set S with respect to a differential system, x' = F(x),  which includes applications to: positivity of solutions of chemical reaction systems in a closed reactor and predator-prey systems, orbital motions in a force field and optimization problems.  A subset S of R³ is said to be a flow invariant set with respect to the differential system x' = F(x), if every solution x= x(t) starting in S (i.e. x(0) in S), remains in S (i.e. x(t) in S), for all  t >0.  Clearly, positivity of all solutions corresponds to  S= R³_+ (the first quadrant).  Using T_xS one can formulate maximum principles of optimum, which contain classical necessary conditions of optimum (like Lagrange multipliers, Euler - Lagrange equation of the curve of most rapid descent etc.).  By introducing the second order tangent cone to S at  x in S, one can study the motion on a given orbit of an object (such as a satellite) in a force field. For the study of a flow invariant set with respect to a semilinear partial differential equations of parabolic type u_t = Delta_x u(t,x) + f(t,u), it is necessary and sufficient to use the notion of the tangency of a vector v to S at x in S in the sense of the semigroup  S(t) generated by Laplace operator Delta in the Sobolev spaces H_o^1(Omega ) intersect H^2(Omega).