Published Paper

Inserted: 19 apr 2007

Journal: SIAM J. Control Optim.
Volume: 44
Number: 1
Pages: 99-110
Year: 2005

Abstract:

We prove the following approximation theorem: given a function $x:[a,b]\to\R^N$ in the Sobolev space $*W*^{\nu+1,1}$, $\nu\geq1$, and $\e>0$, there exists a function $x_{\e}$ in $*W*^{\nu+1,\infty}$ such that $$\begin{array}{c} \displaystyle\int_{a}^{b}\sum_{i=1}^m L_i(x_{\e}^{(\nu)},x_{\e}^{(\nu+1)}) \psi_i(t,x_{\e},x{\e}^{{\prime},\cdots,x}_\e^{(\nu)}) \displaystyle<\int_{a}^{b}\sum_{i=1}^m L_i(x^{(\nu)},x^{(\nu+1)}) \psi_i(t,x,x^{{\prime},\cdots,x{}(\nu)})+\e,

\begin{array}{ll}x_{\e}(a)=x(a),& x_{\e}(b)=x(b), x_{\e}^{\prime}(a)=x^{\prime}(a),&x_{\e}^{\prime}(b)=x^{\prime}(b),

\vdots&

x_\e^{(\nu)}(a)=x^{(\nu)}(a),&x_{\e}^{(\nu)}(b)=x^{(\nu)}(b), \end{array} \end{array}$$ provided that, for every $i$ in $\{1,\cdots,m\}$, $L_i\psi_i$ is continuous in a neighbourhood of $x$, $L_i$ is convex in its second variable and $\psi_i$ evaluated along $x$ has positive sign. We discuss the optimality of our assumptions comparing them with an example of A. V. Sarychev \cite{s}.

As a conseguence, we obtain the non-occurrence of the Lavrentiev phenomenon. In particular, the integral functional $\int_a^b L(x^{(\nu)},x^{(\nu+1)})$ does not exhibit the Lavrentiev phenomenon for any given boundary values $x(a)=A,x(b)=B$, $x^{\prime}(a)=A^{\prime},x^{\prime}(b)=B^\prime$, $\cdots$, $x^{(\nu)}(a)=A^{(\nu)},x^{(\nu)}(b)=B^{(\nu)}$.

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