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Primarni je cilj ovoga rada prikazati osnovne rezultate klasične teorije varijacijskog računa i pripadajuće primjene u problemima ekonomske prirode kroz nekoliko primjera. Navedeni su dovoljni i nužni uvjeti da funkcija \(\bar{x} \in C^1([a,b])\), uz zadane rubne uvjete \(x(t_0)=x_0\) i \(x(t_1)=x_1\) bude maksimizator funkcionala: \(I(x) = \int_{t_0}^{t_1} F \left( t, x(t), x'(t) \right) \, dt,\) pri čemu je funkcija \( F:[a,b] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) neprekidna u svoja 3 argumenta \(t\), \(x\) i \(x'\), i po njima ima neprekidne parcijalne derivacije. Dan je i dodatni uvjet koji maksimizator mora zadovoljavati u slučaju kada je neki od rubnih uvjeta slobodan. Također je dan način za pronalaženje maksimizatora gore navedenog funkcionala u slučaju da je funkcija \(x\) ograničena s \(R(t) \geq x(t), t_0 \leq t \leq t_1\), pri čemu je \(R\) neprekidna funkcija definirana na \([t_0, t_1]\). Predstavljen je još autonomni problem neograničen odozgo te pronalaženje njegovog rješenja metodom najbržeg prilaska. Na kraju su navedeni rezultati primijenjeni na nekoliko problema iz ekonomskog područja, kao na primjer pronalaženje plana proizvodnje koji prati zadani raspored isporuka i minimizira troškove. The primary focus of this thesis is to present some of the basic concepts of the classical theory of the calculus of variations and their applications in economics through several examples. Sufficient and necessary conditions are given so that the function \(\bar{x} \in C^1([a,b])\), with given boundary conditions, \(x(t_0)=x_0\) and \(x(t_1)=x_1\), maximizes the following functional: \(I(x) = \int_{t_0}^{t_1} F \left( t, x(t), x'(t) \right) \, dt,\) is continuous (and has continuous partial derivatives) in its 3 arguments \(t\), \(x\) and \(x'\). An additional condition is given that the maximizing function must satisfy when one of the boundary conditions is free. We have also shown a way to find the maximizing function of the above functional in the case when function \(x\) is bounded by \(R(t) \geq x(t), t_0 \leq t \leq t_1\), where \(R\) is a continuous function defined on \([t_0, t_1]\). An infinite horizon autonomous problem is also presented, as well as finding its solution using the most rapid approach method. Finally, the stated results are applied to several problems in economics, such as finding a production plan that follows a given delivery schedule and minimizes costs. |
