| Abstract | Osnovni cilj ovog rada je upoznati čitatelja s varijacijskim računom neglatkih funkcija. Nakon što smo uveli elementarne pojmove neglatke analize, bili smo spremni riješiti osnovni problem. U varijacijskom računu neglatke funkcije se javljaju kroz dva efekta: neglatka ekstremala \(x\) i neglatka Lagrangeova funkcija \(\Lambda\). Neka je \(x : [a, b] \to \mathbb{R}^{n}\) takav da \(x \in Lip[a, b]. \Lambda(t, x, x') : [a, b] \times \mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}\) predstavlja Lagrangeovu funkciju gdje \(t\) označava vrijeme, \(x\) stanje i \(x'\) brzinu. Tada je osnovni problem varijacijskog računa dan sa min \( J(x) = \int_{a}^{b} \Lambda(t, x(t), x'(t)) dt : x \in Lip[a, b], x(a) = A, x(b) = B. (\textbf{P})\) Glavni rezultat ovog rada, u slučaju neglatkog rješenja \(x_\ast\) je teorem koji nam daje nužan uvjet optimalnosti rješenja : neka je \(x_\ast \in Lip[a, b]\) slabi lokalni minimizator problema (\(\textbf{P}\)), tada \(x_\ast\) zadovoljava integralnu Euler-Lagrangeovu jednadžbu \(\Lambda_v (t, x_\ast(t), x'_\ast(t)) = c + \int_a^t \Lambda_x(s, x_\ast(s), x'_\ast(s) ds, t \in [a, b]\) g.s. gdje je \(c \in \mathbb{R}^n\). U slučaju neglatke Lagrangeove funkcije \(\Lambda\) nužan uvjet optimalnost dan je teoremom : neka je \(\Lambda\) lokalno Lipschitzova funkcija te neka je \(x_\ast \in Lip[a, b]\) rješenje osnovnog problema (\(\textbf{P}\)). Tada postoji Lipschitzova funkcija \(p\) takva da vrijedi \((p'(t), p(t)) \in \partial_C \Lambda(t, x_\ast(t), x'_\ast(t))\) g.s., gdje je \(\partial_C \Lambda\) Clarkeov subdiferencijal s obzirom na \((x, v)\). Kroz nekoliko zanimljivih primjera pokazali smo čitatelju na koje se sve načine koristi varijacijski račun te postavili temelje za daljnje razmatranje teorije varijacijskog računa. |
| Abstract (english) | The main aim of this master thesis is to familiarize the reader with calculus of variations of nonsmooth functions. After introducing the basic concepts of nonsmooth analysis, we were ready to solve the basic problem. In calculus of variations, nonsmooth functions surge through two effects: nonsmooth extremal \(x\) and nonsmooth Lagrangian \(\Lambda\). Let \(x : [a, b] \to \mathbb{R}^{n}\) be such that \(x \in Lip[a, b]. \Lambda(t, x, x') : [a, b] \times \mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}\) presents Lagrangian where \(t\) denotes time, \(x\) state and \(x’\) velocity. Than the basic problem of calculus of variations is given with min \( J(x) = \int_{a}^{b} \Lambda(t, x(t), x'(t)) dt : x \in Lip[a, b], x(a) = A, x(b) = B. (\textbf{P})\) The main result of this thesis in case of nonsmooth extremal \(x_\ast\) is theorem which gives us the necessary condition of the optimality of the solution : let \(x_\ast \in Lip[a, b]\) be a weak local minimizer for the basic problem ( \( \textbf{P}\)), than \(x_\ast\) satisfies the integral Euler-Lagrange equation \(\Lambda_v (t, x_\ast(t), x'_\ast(t)) = c + \int_a^t \Lambda_x(s, x_\ast(s), x'_\ast(s) ds, t \in [a, b] \) g.s. where \(c \in \mathbb{R}^n\). In case of nonsmooth Lagrangian \(\Lambda\), the necessary condition of optimality is given with theorem : let \( \Lambda \) be locally Lipschitz function and let \( x_\ast \in Lip[a, b]\) be solution of the basic problem ( \( \textbf{P} \) ). Than there exists Lipschitz function \(p\) such that \((p'(t), p(t)) \in \partial_C \Lambda(t, x_\ast(t), x'_\ast(t)) \) g.s., where the Clarke subdifferential \(\partial_C \Lambda\) is taken with respect to \((x, v) \). With a few interesting examples we have shown the reader the numerous ways in which calculus of variations can be used and we have set the basis for further consideration of the theory of calculus of variations. |
