The Lagrange method can solve more general problems than(1.1). In particular, the Lagrange function for (3.2)is C=E∑-{f(,n)+入+1(x,,2+1)-+ By the Kuhn- Tucker Theorem, we can again derive the two equations 3.5)and 3.6) References 1] Chiang, A C(1984): Fundamental Methods of Mathematical Economics, McGraw Hill 2 Chiang, AC(1992): Elements of Dymamic Optimization, McGraw-Hill 3 Kamien, M.I.& N.L. Schwartz(1991): Dynamic Optimization, North Holland 4 Sargent, TJ(1987): Dymamic Macroeconomic Theory, Harvard University Press 5 Stokey, N L& R.E. Lucas(1989): Recursive Methods in Economic Dymamics, Har- vard University P. 2. Continuous-Time deterministic model ee Kamien-Schwartz(1991)and Chiang(1992) Example 3.5. The Shortest Path. What is the shortest path to move from one (x1,)inR2? Theorem 3.3. Suppose H: RXR XRK-R is continuous w.r. t. its first argument continuously differentiable w r t. its second and third arguments. Let =[continuously differentiable functions u: [to, T]+RI be the set of admissible controls. Then, the solution u* of u(t), i(t))dtThe Lagrange method can solve more general problems than (1.1). In particular, the Lagrange function for (3.2) is L = Et [∞ s=t βs−t {fs(xs, us) + λs+1 · [gs(xs, us,s+1) − xs+1]} . By the Kuhn-Tucker Theorem, we can again derive the two equations (3.5) and (3.6). References [1] Chiang, A.C. (1984): Fundamental Methods of Mathematical Economics, McGraw￾Hill. [2] Chiang, A.C. (1992): Elements of Dynamic Optimization, McGraw-Hill. [3] Kamien, M.I. & N.L. Schwartz (1991): Dynamic Optimization, North Holland. [4] Sargent, T.J. (1987): Dynamic Macroeconomic Theory, Harvard University Press. [5] Stokey, N.L. & R.E. Lucas (1989): Recursive Methods in Economic Dynamics, Har￾vard University Press. 2. Continuous-Time Deterministic Model See Kamien—Schwartz (1991) and Chiang (1992). Example 3.5. The Shortest Path. What is the shortest path to move from one (x0, y0) to another (x1, y1) in R2?  Theorem 3.3. Suppose H : R×Rk × Rk → R is continuous w.r.t. its first argument, continuously differentiable w.r.t. its second and third arguments. Let A ≡ {continuously differentiable functions u : [t0, T] → Rk} be the set of admissible controls. Then, the solution u∗ of max u∈A ] T t0 H[t, u(t), u˙(t)] dt s.t. u(t0) = u0, u(T) = uT (3.8) 3—4