Page 122 |
Save page Remove page | Previous | 122 of 289 | Next |
|
small (250x250 max)
medium (500x500 max)
Large (1000x1000 max)
Extra Large
large ( > 500x500)
Full Resolution
All (PDF)
|
This page
All
|
if the function '(y, t) =
F
*n D(y, t; x, ' ).(x)dx, satisfies the PDE above and
limt&.− '(y, t) = .(y, ' ).
Before we proceed with the theorem which establishes the connection between the
forward and backward Kolmogorov PDE through the concept of fundamental solution, lets
understand the ”physical” meaning of the function D(y, t; x, ' ) for 0 < t < ', x, y ! "n.
Let us assume for the moment that q(x) = 0 then through the Feynamn Kac lemma
of the solution of the PDE is represented as '(x, t) = E (.(xT )). Inspection of the
last equation and '(y, t) =
F
*n D(y, t; x, ' ).(x)dx tell us that any fundamental solution
of the backward Kolmogotov PDE can be thought as an transition probability of the
stochastic process x which evolves according to the stochastic differential equation (3.9).
Consequently, we can write that D(y, t; x, ') = p(x, ' |y, t). Another property of the
function of fundamental solution of a second order PDE comes from the fact that:
lim
t&.−
'(y, t) = lim
t&.−
"
D(y, t; x, ' ).(x)dx = .(y, ' ) (3.74)
From the equation above, it is easy to see that the fundamental solution has the prop-erty
that limt&.− D(y, t; x, ') = *(y−x) where *(x) is the Dirac function. Clearly, since
there is a probabilistic interpretation of D(y, t; x, ' ) the transition probability p(x, ' |y, t)
inherits the same property and therefore p(x, ' |y, t) = *(y − x) for t = ' .
Now, we present the theorem (Karatzas & Shreve 1991),(Friedman 1975) that estab-lishes
the connection between the forward and backward Kolmogorov PDE, trough the
concept of fundamental solution.
108
Object Description
| Title | Iterative path integral stochastic optimal control: theory and applications to motor control |
| Author | Theodorou, Evangelos A. |
| Author email | etheodor@usc.edu; theo0027@umn.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Computer Science |
| School | Viterbi School of Engineering |
| Date defended/completed | 2011-01-11 |
| Date submitted | 2011 |
| Restricted until | Unrestricted |
| Date published | 2011-04-29 |
| Advisor (committee chair) | Schaal, Stefan |
| Advisor (committee member) |
Valero-Cuevas, Francisco Sukhatme, Gaurav S. Todorov, Emo Schweighofer, Nicolas |
| Abstract | Motivated by the limitations of current optimal control and reinforcement learning methods in terms of their efficiency and scalability, this thesis proposes an iterative stochastic optimal control approach based on the generalized path integral formalism. More precisely, we suggest the use of the framework of stochastic optimal control with path integrals to derive a novel approach to RL with parameterized policies. While solidly grounded in value function estimation and optimal control based on the stochastic Hamilton Jacobi Bellman (HJB) equation, policy improvements can be transformed into an approximation problem of a path integral which has no open algorithmic parameters other than the exploration noise. The resulting algorithm can be conceived of as model-based, semi-model-based, or even model free, depending on how the learning problem is structured. The new algorithm, Policy Improvement with Path Integrals (PI2), demonstrates interesting similarities with previous RL research in the framework of probability matching and provides intuition why the slightly heuristically motivated probability matching approach can actually perform well. Applications to high dimensional robotic systems are presented for a variety of tasks that require optimal planning and gain scheduling.; In addition to the work on generalized path integral stochastic optimal control, in this thesis we extend model based iterative optimal control algorithms to the stochastic setting. More precisely we derive the Differential Dynamic Programming algorithm for stochastic systems with state and control multiplicative noise. Finally, in the last part of this thesis, model based iterative optimal control methods are applied to bio-mechanical models of the index finger with the goal to find the underlying tendon forces applied for the movements of, tapping and flexing. |
| Keyword | stochastic optimal control; reinforcement learning,; robotics |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m3804 |
| Contributing entity | University of Southern California |
| Rights | Theodorou, Evangelos A. |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | cisadmin@lib.usc.edu |
| Filename | etd-Theodorou-4581 |
| Archival file | uscthesesreloadpub_Volume14/etd-Theodorou-4581.pdf |
Description
| Title | Page 122 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | if the function '(y, t) = F *n D(y, t; x, ' ).(x)dx, satisfies the PDE above and limt&.− '(y, t) = .(y, ' ). Before we proceed with the theorem which establishes the connection between the forward and backward Kolmogorov PDE through the concept of fundamental solution, lets understand the ”physical” meaning of the function D(y, t; x, ' ) for 0 < t < ', x, y ! "n. Let us assume for the moment that q(x) = 0 then through the Feynamn Kac lemma of the solution of the PDE is represented as '(x, t) = E (.(xT )). Inspection of the last equation and '(y, t) = F *n D(y, t; x, ' ).(x)dx tell us that any fundamental solution of the backward Kolmogotov PDE can be thought as an transition probability of the stochastic process x which evolves according to the stochastic differential equation (3.9). Consequently, we can write that D(y, t; x, ') = p(x, ' |y, t). Another property of the function of fundamental solution of a second order PDE comes from the fact that: lim t&.− '(y, t) = lim t&.− " D(y, t; x, ' ).(x)dx = .(y, ' ) (3.74) From the equation above, it is easy to see that the fundamental solution has the prop-erty that limt&.− D(y, t; x, ') = *(y−x) where *(x) is the Dirac function. Clearly, since there is a probabilistic interpretation of D(y, t; x, ' ) the transition probability p(x, ' |y, t) inherits the same property and therefore p(x, ' |y, t) = *(y − x) for t = ' . Now, we present the theorem (Karatzas & Shreve 1991),(Friedman 1975) that estab-lishes the connection between the forward and backward Kolmogorov PDE, trough the concept of fundamental solution. 108 |
