Homework assignments

MATH 513 - Homework 10
(due Thursday, Dec. 11 2014)
1. (6pts.) Consider the optimization problem in infinite time horizon:
Z ∞
cos x(t)e−t dt
maximize
0
subject to
x(t)
˙
= u(t) = [−1, 1] ,
x(0) = y .
Call V the optimal value function. In other words, V (y) is the maximum discounted payoff
that can be achieved if the initial state is x(0) = y.
(i) Write the Hamilton-Jacobi PDE satisfied by the function V (this is actually an ODE,
because x ∈ IR).
(ii) Find the optimal control u∗ (x). Note: this control should steer the system toward points
where the running payoff is maximum: it is easy to guess what it should be.
(iii) Using (ii), explicitly compute the value function V (y).
Hint: By periodicity, it is enough to compute it on [0, 2π]. To explicitly compute the
d e−t (sin t−cos t)
total cost by an integration, note that dt
= e−t cos t.
2
(iv) Check that the value function provides a viscosity solution to the Hamilton-Jacobi PDE.
2. (4 pts.) Consider the conservation law
ut +
u3
3
= 0.
x
(i) Decide which of the following piecewise constant functions are weak solutions. Are they
entropy-admissible?
1 if x < t/3
u1 (t, x) =
−1 if x > t/3
0 if x < t
u3 (t, x) =
3 if x > t
u2 (t, x) =
u4 (t, x) =
1
0
−1 if x < t/3
0 if x > t/3
if x < t/3
if x > t/3
(ii) In the t-x plane, sketch the entropy-admissible solution to the Cauchy problem with
initial data
1 if x < 0
u(0, x) =
−1 if x > 0
1
MATH 513 - Homework 9
(due Tuesday, Dec. 2, 2014)
1. (3pts.) Let V = V (t, x) be a continuous function that provides a viscosity solution to the
Hamilton-Jacobi equation
Vt + H(x, ∇V ) = 0.
Write the equation satisfied by W (t, x) = −V (t, x), in the viscosity sense. Justify your answer.
2. (3pts.) Let W = W (τ, y) be the value function for the following optimal control problem
with running cost and terminal payoff:
Z
maximize:
T
ψ(x(T )) −
L(x(t), u(t)) dt ,
τ
subject to
x(t)
˙
= f (x(t), u(t)) ,
x(τ ) = y,
u(t) ∈ U .
Write a PDE satisfied by W , in the viscosity sense.
(Hint: you can transform the above into a minimization problem, and use the results proved
in class).
3. (4pts.) Consider a forced pendulum, governed by
x
¨(t) + sin x(t) = u(t).
(1)
Here t 7→ u(t) ∈ [0, 1] is an external force, which we use as a control. For a given initial data
x(τ ) = y1 ,
x(τ
˙ ) = y2
consider the optimization problem:
minimize:
x2 (T ) x˙ 2 (T )
+
.
2
2
Write a PDE satisfied by the value function V = V (τ, y1 , y2 ) in the viscosity sense, together
with appropriate boundary values.
Knowing V , write a formula for the optimal control u = u∗ (τ, y1 , y2 ). Observe that, in general,
this will be discontinuous. On some regions, the value u = 1 will be optimal, on other regions,
the value u = 0 will be optimal.
(Hint: start by writing the second order ODE (1) as a first order system).
2
MATH 513 - Homework 8
(due Tuesday, Nov.18, 2014)
1. (3pts.) Let Ω ⊂ IRn be a bounded open set. Let u1 , u2 be two viscosity solutions of
 3
x ∈ Ω,
 u + |∇u| − 1 = 0
x ∈ ∂Ω .
u = 0

Assuming that u1 is continuous while u2 is continuously differentiable, prove that u1 = u2 .
2. (3pts.) In the x1 -x2 plane, consider the open rectangle
n
o
.
Ω = (x1 , x2 ) ; |x1 | < 2 , |x2 | < 1 .
Show that the function
x ∈ Ω,
u(x) = dist(x, ∂Ω)
measuring the distance of a point x ∈ Ω to the boundary, is a viscosity solution of

x ∈ Ω,
 |∇u|2 − 1 = 0

u = 0
x ∈ ∂Ω .
(Write out explicitly the function u. Take advantage of symmetries to reduce your computations.)
2. (4pts.) For t0 < T , consider the optimal control problem
.
minimize: J(t0 , x0 , u) = x(T ) +
Z
T
[x2 (t) + 4u2 (t)] dt ,
t0
subject to
x(t0 ) = x0 ,
x(t)
˙
= x(t) + u(t),
u(t) ∈ IR.
• Write a PDE satisfied by the value function:
.
V (t0 , x0 ) = min J(t0 , x0 ; u),
u(·)
together with suitable boundary conditions.
• Write a formula providing the optimal feedback control u(x), assuming that the function
V is known.
3
MATH 513 - Homework 7
(due Thursday, Oct. 30, 2014)
1. (3pts.) Find a solution to the first order PDE
u2t + u2x = 4 ,
(1)
u(0, x) = sin x .
(2)
with initial data at t = 0
Sketch the characteristic curves. Find the maximal time interval t ∈ [0, T ] on which the
solution remains smooth. Locate the points where the regularity first breaks down: |ux | → ∞.
2. (3pts.) Find a solution to the PDE
ut + uux = 0
(3)
with initial data (2). Sketch the characteristic curves. Find the maximal time interval t ∈ [0, T ]
on which the solution remains smooth. Locate the points where the regularity breaks down.
3. (4pts.) For the following PDEs, write the equations of characteristics.
(i)
ut + xux = sin u,
(ii)
ut − uux = 0,
(iii)
ut + ux = u2 .
Given the initial data
1
,
1 + x2
in which cases can we say that a globally defined smooth solution exists?
u(x, 0) =
4
MATH 513 - Homework 6
(due Thursday, Oct. 16, 2014)
1. (4 pts.) Let u ∈ C 2 (IR × [0, ∞[ ) be a solution of the wave equation in one space dimension:
utt − uxx = 0
in IR× ]0, ∞[ ,
and assume that the initial data
u(x, 0) = g(x),
ut (x, 0) = h(x)
have bounded support. Define the kinetic energy K(t) and the potential energy P (t) as
Z ∞ 2
Z ∞ 2
ux
ut
.
.
dx ,
P (t) =
dx .
K(t) =
−∞ 2
−∞ 2
Prove that
(i) E(t) = K(t) + P (t) is constant for all times t,
(ii) There exists a time T sufficiently large so that K(t) = P (t) for all t > T .
Hint: assume g(x) = h(x) = 0 for |x| ≥ R. Check what happens to the solution for t > 2R.
2. (3 pts.) Consider the wave equation with additional elastic source
utt − c2 uxx + u = 0,
with smooth initial data
u(x, 0) = g(x),
ut (x, 0) = h(x) .
Using an energy method, prove that the Cauchy problem can have at most one solution.
3. (3 pts.) Let u ∈ C 2 (IR3 × [0, ∞[ ) be a solution to the wave equation in a three dimensional
space:
utt − ∆uxx = 0
in IR3 × ]0, ∞[ ,
and assume that the initial data
u(x, 0) = g(x),
ut (x, 0) = h(x)
has compact support, say g(x) = h(x) = 0 for |x| ≥ R. prove that there exists a constant C
such that
C
x ∈ IR3 , t > 0 .
|u(x, t)| ≤
t
5
MATH 513 - Homework 5
(due Thursday, Oct. 9, 2014)
1. (4 pts.) We say that v ∈ C 2 (Ω) is a subsolution of the heat equation if
.
in ΩT = Ω×]0, T [ .
vt − ∆v ≤ 0
(i) Let u be a solution and let v be a subsolution of the heat equation, such that u, v are
continuous on the closure ΩT and v ≤ u one the parabolic boundary ΓT of ΩT .
Prove that v ≤ u on the entire domain ΩT .
(ii) Let φ : IR →
7 IR be smooth and convex. If u solves the heat equation and v = φ(u),
prove that v is a subsolution.
2. (3 pts.) On the Euclidean space IRn with variable ξ = (ξ1 , . . . , ξn ), consider the gradient
flow
n
X
k2 2
ξ˙ = −∇Φ(ξ),
Φ(ξ) =
ξ .
(1)
2 k
k=1
(i) Explicitly write the corresponding system of ODEs for (ξ1 , . . . , ξn )
P
(ii) Prove that the function u(x, t) = nk=1 ξk (t) sin kx is a solution of the heat equation

t > 0, x ∈ ]0, π[ ,
 ut − ∆u = 0 ,

u(0, t) = u(π, t) = 0
t≥0
if and only if t 7→ ξ(t) = (ξ1 (t), . . . , ξn (t)) is a solution of the gradient flow (1).
3. (3 pts.)
(i) Show that the general solution to the PDE
uXY = 0
is
u(X, Y ) = F (X) + G(Y ).
(ii) Performing the change of variables X = x + t, Y = x − t, show that
utt − uxx = 0
if and only if
uXY = 0.
(iii) Using the above, give an alternative derivation of D’Alembert’s formula.
6
MATH 513 - Homework 4
(due Thursday, Oct. 2, 2014)
1. (3 pts.) Let u be a smooth solutions of the heat equation ut − ∆u = 0 in IRn × ]0, ∞[ .
(i) Prove that uλ (x, t) = u(λx, λ2 t) is another solution of the heat equation, for every λ ∈ IR.
(ii) Using (i), prove that v(x, t) = x · Du(x, t) + 2tut (x, t) solves the heat equation as well.
2. (3 pts.) Write an explicit formula for the solution to the non-homogeneous problem
ut − ∆u + cu = f (x, t)
(x, t) ∈ IRn × ]0, ∞[ ,
u(x, 0) = g(x)
x ∈ IRn .
Here c ∈ IR. Hint: write an equation satisfied by the function v(x, t) = ect u(x, t).
3. (2 pts.) Let p(x) = a0 + a1 x + · · · + aN xN be any polynomial. Prove that the solution to
the one-dimensional heat equation
ut = uxx
u(x, 0) = p(x)
is given by the exponential formula
u(t, x) =
∞ k
X
t ∂ 2k
p(x) .
k! ∂x2k
k=0
Notice that only finitely many terms are 6= 0, hence the series trivially converges. Formally,
the right hand side can be interpreted as et∆ p.
4. (2 pts.) Let Ω ⊂ IRn be a bounded open set, call ΩT = Ω× ]0, T [ , and let ΓT be the
parabolic boundary of ΩT .
Let u be a solution to

 ut = ∆u + f

u =g
(x, t) ∈ ΩT ,
(x, t) ∈ ΓT .
Assume f (x, t) ≤ K, g(x, t) ≤ M . Prove that u(x, t) ≤ K + tM for all (x, t) ∈ ΩT .
7
MATH 513 - Homework 3
(due Thursday, Sept. 25, 2014)
1 (4pts). Consider the unit circle Ω = {(x1 , x2 ) ∈ IR2 ; x21 + x22 < 1}. Write an integral
formula for the solution to the boundary value problem
∆u = 0
x ∈ Ω,
u(x1 , x2 ) = x21
x ∈ ∂Ω.
Explicitly compute u(0, 0), i.e. the value of the solution at the origin.
(Hint: use polar coordinates).
2 (2+2 pts). Consider the half plane Ω = {(x1 , x2 ) ∈ IR2 ; x2 > 0}.
(i) Write an integral formula for the solution of
∆u = 0
u(x1 , 0) = g(x1 ),
x ∈ Ω,
with g bounded and continuous, g(x1 ) = |x1 | for |x1 | ≤ 1.
(ii) Show that the gradient Du is not bounded near the origin.
(Hint: estimate u(0,h)−u(0,0)
as h → 0+).
h
3 (2pts). On the bounded open set Ω ⊂ IR2 , consider the problem
∆u = x1 sin x2
x ∈ Ω,
u(x1 , x2 ) = ex1
x ∈ ∂Ω.
We wish to characterize the solution u as a minimizer:
I[u] = min I[w]
w∈A
Write out a suitable integrand I[w] and the set A.
8
MATH 513 - Homework 2
(due Tuesday, Sept. 16, 2014)
1 (4pts). Prove that, for every smooth function φ : IR3 7→ IR with compact support, one has
Z
1
∆φ(x) dx = C · φ(0) .
R3 |x|
Explicitly compute the constant C.
2 (3pts). We say that a function v ∈ C 2 is subharmonic if it satisfies the inequality −∆v(x) ≤ 0
at every point x of its domain.
Let now u : IRn →
7 IR be a harmonic function and let φ : IR 7→ IR be a smooth convex function,
.
so that φ00 ≥ 0. Prove that w(x) = φ(u(x)) is subharmonic.
3 (3pts). Let v be a subharmonic function on an open, bounded domain Ω, with v continuous
on the closure Ω.
Show that, for every ball B(x, r) contained in Ω, one has
Z
.
v(x) ≤ −−
v(y) dy = [average value of v on the ball B(x, r)]
B(x,r)
Deduce that
max v(x) = max v(x) .
x∈Ω
x∈∂Ω
9
MATH 513 - Homework 1
(due Tuesday, Sept. 9, 2014)
1. (3 pts) Consider the function u(x, t) = a ebx+ct . Find for which values of the constants
a, b, c ∈ IR is this a solution
(i) of the heat equation: ut = uxx
(ii) of the wave equation: utt = uxx
(iii) of the Laplace equation: utt + uxx = 0
2. (2 pts) Classify the following PDEs
(i)
ut + uxxx = 3u + sin x
(ii)
utt − (eu ux )x = 0
(iii)
u2x + u2y = x2 + y 2
(iv)
ut + 3ux = u2 + uxx
3. (2 pts) Find a function u(t, x) such that
ut − ux = x2 + t ,
u(0, x) = cos x .
4. (3 pts) Let u be a solution to the Laplace equation:
x ∈ IRn .
∆u = 0
Let R = (Rij ) be an n × n orthogonal matrix, so that RRT = RT R = I, where RT is the
transpose of R and I denotes the identity matrix. Prove that v(x) = u(Rx) is another solution
to the Laplace equation.
P
Hint: consider the additional variable y = Rx, so that yi =
j Rij xj and v(x) = u(y).
Compute ∆v using the chain rule.
10

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