In Harry Potter and the Order of the Phoenix, Umbridge's use of the Dementors that she sent to Privet Drive makes no sense. She should have made sure that Dudley wasn't the only Muggle there. At the hearing, Fudge says that Harry performed "magic in the presence of a Muggle". Note the singular: this was a Muggle who was already aware of the wizarding world.
Why didn't Umbridge make it so that Harry was forced to use magic in front of someone other than Dudley?
Let's look at Umbridge's own statement about why she sent the Dementors in the first place.
"It was you?" gasped Harry. "You sent the Dementors after me?"
"Somebody had to act," breathed Umbridge, as her wand came to rest pointing directly at Harry's forehead. "They were all bleating about about silencing you somehow - discrediting you - but I was the only one who actually did something about it..."
(Order of the Phoenix, Chapter 32, Out of the Fire).
So her intention was to silence and/or discredit Harry. How did she plan to do that? As Janus Bahs Jacquet says, the primary goal was to have the Dementors administer the Kiss, thereby sucking out Harry's soul and leaving him in a vegetative state. This would be the easiest way of "silencing" him.
It may have occurred to Umbridge that Harry, being the resourceful and talented wizard that he's known to be, may have survived the attack. Of course, he would have to use magic outside of school in order to do so. Umbridge likely looked up Harry's file and saw that he already had a warning for underage magic. Thus she concluded that, should Harry survive, he would duly be expelled and marked as a criminal renegade in the eyes of the wizarding community. This would be a means of effectively "discrediting" Harry.
As it happened, Harry ended up facing a trial by the Wizengamot. Dudley's presence is not strictly relevant here, or at least not essential for Harry to be expelled.
He extricated a piece of parchment from the pile before him, took a deep breath, and read out, "The charges against the accused are as follows:
"That he did knowingly, deliberately and in full awareness of the illegality of his actions, having received a previous written warning from the Ministry of Magic on a similar charge, produce a Patronus Charm in a Muggle-inhabited area, in the presence of a Muggle, on the second of August at twenty-three minutes past nine, which constitutes an offence under Paragraph C of the Decree for the Reasonable Restriction of Underage Sorcery, 1875, and also under Section 13 of the International Confederation of Warlocks' Statute of Secrecy."
(Order of the Phoenix, Chapter 8, The Hearing).
There are three things to note.
Firstly, as far as the Ministry was concerned, it doesn't matter whether or not Dudley knows about magic. From their point of view, a Muggle is a Muggle is a Muggle. A very small number of Muggles (the Prime Minister and those related to wizards and witches) may know about magic. But performing magic in front of them still counts, strictly speaking, as a violation of the Statute of Secrecy. Remember that Fudge wasn't interested in giving Harry a fair trial. If there were mitigating factors which proved Harry's innocence Fudge was very happy to ignore them.
Secondly, the charge under the Statute of Secrecy wasn't just that Harry performed magic in front of Dudley but that he did it in a "Muggle-inhabited area". Fudge's point was that anyone could have seen Harry's magic, not just Dudley - especially as the Patronus Charm, when producing a fully corporeal Patronus, is a particularly bright and attention-grabbing spell. Dudley didn't have to have been there for Harry to have broken the law.
Thirdly, even if Harry was innocent under the second charge he still could've been guilty under the first. He was being accused of underage magic as well of breaking the Statute of Secrecy. And this was the stronger charge of the two since Harry had already received a written warning. Harry could've been alone that night and still have been guilty of doing magic outside of school in the eyes of the law.
Dudley wasn't needed by Umbridge to force Harry to break the Reasonable Restriction of Underage Sorcery. She probably didn't even know that he was going to be there. His presence didn't hurt her plan but it wasn't an essential part of it.
