VariationalCalculus - Maple Programming Help

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VariationalCalculus

 Convex
 determine whether an integrand is convex

 Calling Sequence Convex(f, t, x(t))

Parameters

 f - expression in t, x(t), and x'(t) t - independent variable x(t) - unknown function (or list of functions)

Description

 • The Convex(f, t, x(t)) command determines if the integrand is convex.
 • If the integrand is convex, the functional $J=\underset{a}{\overset{b}{\int }}f\left(t,x,x,'\right)ⅆt$ is globally minimized by extremals (solutions of the Euler-Lagrange equations).
 • For a convex integrand, the output is an expression sequence containing two items:
 – Hessian matrix $\frac{{\partial }^{2}}{\partial x\partial x\text{'}}f$
 – Logical expression that is true iff the Hessian is positive semidefinite, which proves that J is a minimum
 • If the integrand is not convex, Maple returns false.
 • If LinearAlgebra[IsDefinite] cannot determine the convexity, the output is an expression sequence containing two items:
 – Hessian matrix $\frac{{\partial }^{2}}{\partial x\partial x\text{'}}f$
 – unevaluated call to IsDefinite
 • If an error occurs in the execution of LinearAlgebra[IsDefinite], only the Hessian matrix is returned.
 • The arithmetic negation makes the Hessian negative semidefinite.

Examples

 > $\mathrm{with}\left(\mathrm{VariationalCalculus}\right)$
 $\left[{\mathrm{ConjugateEquation}}{,}{\mathrm{Convex}}{,}{\mathrm{EulerLagrange}}{,}{\mathrm{Jacobi}}{,}{\mathrm{Weierstrass}}\right]$ (1)
 > $f≔{\left({\left(\frac{ⅆ}{ⅆt}x\left(t\right)\right)}^{2}+{\left(\frac{ⅆ}{ⅆt}y\left(t\right)\right)}^{2}\right)}^{\frac{1}{2}}$
 ${f}{≔}\sqrt{{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}^{{2}}{+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}}$ (2)
 > $\mathrm{Convex}\left(f,t,\left[x\left(t\right),y\left(t\right)\right]\right)$
 $\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {-}\frac{{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}^{{2}}}{{\left({\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}^{{2}}{+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}\right)}^{{3}}{{2}}}}{+}\frac{{1}}{\sqrt{{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}^{{2}}{+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}}}& {-}\frac{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}{{\left({\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}^{{2}}{+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}\right)}^{{3}}{{2}}}}\\ {0}& {0}& {-}\frac{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}{{\left({\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}^{{2}}{+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}\right)}^{{3}}{{2}}}}& {-}\frac{{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}}{{\left({\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}^{{2}}{+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}\right)}^{{3}}{{2}}}}{+}\frac{{1}}{\sqrt{{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}^{{2}}{+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}}}\end{array}\right]{,}{0}{\le }\frac{{1}}{\sqrt{{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}^{{2}}{+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}}}$ (3)