Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera, D. Patiño Departamento de Matemáticas Universidad de los Andes Colombia, 2005 SIAM Conference on Optimization Stockholm, May 2005

Introduction Analysis of the problem Problem of the moments Examples Index Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Introduction We will analyze the variational problem: The integrand f is given by the general expression: f is a polynomial in the derivative y’ and  and  are integrable: Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Introduction We use Young measure:And the link between y and v is given by: Then we can use the moment theory for the parametrized measure and the boundary conditions The task is to determine a probability measures: Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Analysis of the problem We take the original problem and we transform it into a problem including its derivative constraints. The new integrand is: We may use the one dimensional Solobev space H 1,p (a,b) as the family of admissible functions where p is the degree of the polynomial. A is the set of values (x,y, ) satisfying the inequalities Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Analysis of the problem We use the Direct Method if the Integrand is coercive and semi- continuous functional: We assume for the problem: Every minimizing sequence u n has a weakly convergent subsequence and we characterize the weak limit of the sequence: Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Analysis of the problem We can determine minimizing sequences for the functional I by solving the problem: The measure could be composed by one delta: Or two deltas: Then: Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Analysis of the problem Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera Then the problem can be relaxed: The generalized functional satisfies: The support of every optimal parametrized measure must be contained in the set of points satisfies: 3. The problem has a minimizer if the Young measures is composed of Dirac measures:

Problem of the moments Problem of the moments of Hausdorff: Problem of the moments of Hausdorff: We characterize the moments of a probability measure supported in the interval [l,  ) We define de cone of the moments of measures in the interval [a,b]: And the cone of the positives function in [a,b] as: Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Problem of the moments Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera The vector [m 0, …,m 2n ] is in the closure of the cone of algebraic moments of the positive measure in [a,b]. And the following matrix are positive semidefinites.

Problem of the moments Theorem of relaxation: We use this theorem for changing the problem and characterize the measure with moments. We transform the problem in a conical program: Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Problem of the moments Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera Measure: If m 2 (x)=x 2, then: If m 2 (x)>x 2, the values of a 1 (x), a 2 (x): The weights of the probability measure are:

Example 1 We take the problem: Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Example 1 a=0, b=1,  =0.5 a=b=0  =0 Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Example 1 a=0.5, b=0,  =0.5 Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Example 2 We change the problem to: a=b=0  =-0.1 Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Example 2 a=b=0  =-x/2 Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Example 3 We take the problem: Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Example 3 a=0 b=0.1  =0 Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera

Example 4 We analyse the problem: Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera a=0 b=0.1  =-0.5  =0.5 g(t)=t 2

Example 4 Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera a=0.5 b=0.2  =-0.5  =0.5 g(t)=t/4

Example 5 Non convex variational problems under restrictions on the derivative Non convex variational problems under restrictions on the derivative R. Meziat, C. Rivera We take the problem: a=0 b=0  =0.2